Add and Subtract Polynomials

Determine the Degree of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form \(a{x}^{m},\) where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomials in one variable are \(2x,5y,17z,\) and \(4{y}^{2}\). Monomials can also have more than one variable such as \(5abc\) and \(-4{a}^{2}{b}^{3}{c}^{2}.\)

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Here are some examples of polynomials.

Polynomial	\(y+1\)	\(4{a}^{2}-7ab+2{b}^{2}\)	\(4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1\)
Monomial	14	\(8{y}^{2}\)	\(-9{x}^{3}{y}^{5}\)	\(-13{a}^{3}{b}^{2}c\)
Binomial	\(a+7b\)	\(4{x}^{2}-{y}^{2}\)	\({y}^{2}-16\)	\(3{p}^{3}q-9{p}^{2}q\)
Trinomial	\({x}^{2}-7x+12\)	\(9{m}^{2}+2mn-8{n}^{2}\)	\(6{k}^{4}-{k}^{3}+8k\)	\({z}^{4}+3{z}^{2}-1\)

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example

Try it.

Add or subtract: ⓐ \(25{y}^{2}+15{y}^{2}\) ⓑ \(16p{q}^{3}-(-7p{q}^{3}).\)

Solution

ⓐ
\(\begin{array}{llll} & & & \ 25{y}^{2}+15{y}^{2} \\ \text{Combine like terms.} & & & \ 40{y}^{2}\end{array}\)

ⓑ
\(\begin{array}{llll} & & & \ 16p{q}^{3}-(-7p{q}^{3}) \\ \text{Combine like terms.} & & & \ 23p{q}^{3}\end{array}\)

Remember that like terms must have the same variables with the same exponents.

Example

Try it.

Simplify: ⓐ \({a}^{2}+7{b}^{2}-6{a}^{2}\) ⓑ \({u}^{2}v+5{u}^{2}-3{v}^{2}.\)

Solution

ⓐ

	\(\ {a}^{2}+7{b}^{2}-6{a}^{2}\)
Combine like terms.	\(\ -5{a}^{2}+7{b}^{2}\)

ⓑ

	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)
There are no like terms to combine. In this case, the polynomial is unchanged.	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms, those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example

Try it.

Find the sum:\((7{y}^{2}-2y+9)+(4{y}^{2}-8y-7).\)

Solution

Identify like terms.	\((\underline{\underline{7{y}^{2}}}-\underline{2y}+9)+(\underline{\underline{4{y}^{2}}}-\underline{8y}-7)\)
Rewrite without the parentheses, rearranging to get the like terms together.	\(\underline{\underline{7{y}^{2}+4{y}^{2}}}-\underline{2y-8y}+9-7\)
Combine like terms.	\(11{y}^{2}-10y+2\)

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

To subtract \(a\) from \(b,\) we write it as \(b-a,\) placing the \(b\) first.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, \(f(x)={x}^{2}+5x+6\) and \(g(x)=3x-4\) are polynomial functions, because \({x}^{2}+5x+6\) and \(3x-4\) are polynomials.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of \(f(x)\) for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example

Try it.

For the function \(f(x)=5{x}^{2}-8x+4\) find: ⓐ \(f(4)\) ⓑ \(f(-2)\) ⓒ \(f(0).\)

Solution

ⓐ

Simplify the exponents.
Multiply.
Simplify.

ⓑ

Simplify the exponents.
Multiply.
Simplify.

ⓒ

Simplify the exponents.
Multiply.
Simplify.

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example

Try it.

The polynomial function \(h(t)=-16{t}^{2}+250\) gives the height of a ball t seconds after it is dropped from a 250-foot tall building. Find the height after \(t=2\) seconds.

Solution

	\(h(t)=-16{t}^{2}+250\)
To find \(h(2),\) substitute \(t=2.\)	\(h(2)=-16{(2)}^{2}+250\)
Simplify.	\(h(2)=-16\cdot 4+250\)
Simplify.	\(h(2)=-64+250\)
Simplify.	\(h(2)=186\)
	After 2 seconds the height of the ball is 186 feet.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Example

Try it.

For functions \(f(x)=3{x}^{2}-5x+7\) and \(g(x)={x}^{2}-4x-3,\) find:

ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

Solution

ⓐ

Rewrite without the parentheses.
Put like terms together.
Combine like terms.

ⓑ In part (a) we found \((f+g)(x)\) and now are asked to find \((f+g)(3).\)

	\((f+g)(x)=4{x}^{2}-9x+4\)
To find \((f+g)(3),\) substitute \(x=3.\)	\((f+g)(3)=4{(3)}^{2}-9\cdot 3+4\)
	\((f+g)(3)=4\cdot 9-9\cdot 3+4\)
	\((f+g)(3)=36-27+4\)

Notice that we could have found \((f+g)(3)\) by first finding the values of \(f(3)\) and \(g(3)\) separately and then adding the results.

Find \(f(3).\)
Find \(g(3).\)
Find \((f+g)(3).\)
\(\)

ⓒ

Rewrite without the parentheses.
Put like terms together.
Combine like terms.

ⓓ

Key Concepts

• Monomial
  • A monomial is an algebraic expression with one term.
  • A monomial in one variable is a term of the form \(a{x}^{m},\) where a is a constant and m is a whole number.
• Polynomials
  • Polynomial, A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
  • monomial, A polynomial with exactly one term is called a monomial.
  • binomial, A polynomial with exactly two terms is called a binomial.
  • trinomial, A polynomial with exactly three terms is called a trinomial.
• Degree of a Polynomial
  • The degree of a term is the sum of the exponents of its variables.
  • The degree of a constant is 0.
  • The degree of a polynomial is the highest degree of all its terms.

Add and Subtract Polynomials

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

Try it.

ⓐ \(47{x}^{5}-17{x}^{2}{y}^{3}+{y}^{2}\)
ⓑ \(5{c}^{3}+11{c}^{2}-c-8\)
ⓒ \(\frac{5}{9}ab+\frac{1}{3}b\)
ⓓ 4
ⓔ \(4pq+17\)

Solution

ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
ⓔ binomial, 2

Try it.

ⓐ \({x}^{2}-{y}^{2}\)
ⓑ \(-13{c}^{4}\)
ⓒ \({a}^{2}+2ab-7{b}^{2}\)
ⓓ \(4{x}^{2}{y}^{2}-3xy+8\)
ⓔ 19

Try it.

ⓐ \(8y-5x\)
ⓑ \({y}^{2}-5yz-6{z}^{2}\)
ⓒ \({y}^{3}-8{y}^{2}+2y-16\)
ⓓ \(81a{b}^{4}-24{a}^{2}{b}^{2}+3b\)
ⓔ \(-18\)

Solution

ⓐ binomial, 1ⓑ trinomial, 2
ⓒ polynomial, 3ⓓ trinomial, 5
ⓔ monomial, 0

Try it.

ⓐ \(11{y}^{2}\)
ⓑ \(-73\)
ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\)
ⓓ \(4{y}^{2}+17{z}^{2}\)
ⓔ \(5{c}^{3}+11{c}^{2}-c-8\)

Try it.

ⓐ \(5{a}^{2}+12ab-7{b}^{2}\)
ⓑ \(18x{y}^{2}z\)
ⓒ \(5x+2\)
ⓓ \({y}^{3}-8{y}^{2}+2y-16\)
ⓔ \(-24\)

Solution

ⓐ \(\text{trinomial, }2\) ⓑ \(\text{monomial, }4\) ⓒ \(\text{binomial, }1\) ⓓ \(\text{polynomial, }3\)
ⓔ \(\text{monomial, }0\)

Try it.

ⓐ \(9{y}^{3}-10{y}^{2}+2y-6\)
ⓑ \(-12{p}^{3}q\)
ⓒ \({a}^{2}+9ab+18{b}^{2}\)
ⓓ \(20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30\)
ⓔ 17

Try it.

ⓐ \(14s-29t\)
ⓑ \({z}^{2}-5z-6\)
ⓒ \({y}^{3}-8{y}^{2}z+2y{z}^{2}-16{z}^{3}\)
ⓓ \(23a{b}^{2}-14\)
ⓔ \(-3\)

Solution

ⓐ \(\text{binomial, }1\) ⓑ \(\text{trinomial, }2\) ⓒ \(\text{polynomial, }3\) ⓓ \(\text{binomial, }3\)
ⓔ \(\text{monomial, }0\)

Try it.

ⓐ \(15x{y}^{}\)
ⓑ 15
ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\)
ⓓ \(10p-9q\)
ⓔ \({m}^{4}+4{m}^{3}+6{m}^{2}+4m+1\)

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

Try it.

ⓐ \({\ \text{7x}}^{\text{2}}+5{x}^{2}\)
ⓑ \(\ \text{4a}-9a\)

Solution

ⓐ \({\text{12x}}^{\text{2}}\) ⓑ \(\text{-}\ \text{5a}\)

Try it.

ⓐ \({\ \text{4y}}^{\text{3}}+6{y}^{3}\)
ⓑ \(\text{-}y-5y\)

Try it.

ⓐ \(-12w+18w\)
ⓑ \(7{x}^{2}y-(-12{x}^{2}y)\)

Solution

ⓐ \(\text{6}w\) ⓑ \(19{x}^{2}y\)

Try it.

ⓐ \(-3m+9m\)
ⓑ \(15y{z}^{2}-(-8y{z}^{2})\)

Try it.

\({\ \text{7x}}^{\text{2}}+5{x}^{2}+\ \text{4a}-9a\)

Solution

\({\text{12x}}^{\text{2}}-\ \text{5a}\)

Try it.

\({\ \text{4y}}^{\text{3}}+6{y}^{3}-y-5y\)

Try it.

\(-12w+18w+7{x}^{2}y-(-12{x}^{2}y)\)

Solution

\(6w+19{x}^{2}y\)

Try it.

\(-3m+9m+15y{z}^{2}-(-8y{z}^{2})\)

Try it.

ⓐ \(-5b-17b\)
ⓑ \(3xy-(-8xy)+5xy\)

Solution

ⓐ \(-22b\) ⓑ \(16xy\)

Try it.

ⓐ \(-10x-35x\)
ⓑ \(17m{n}^{2}-(-9m{n}^{2})+3m{n}^{2}\)

Try it.

ⓐ \(\ \text{12}a+5b-22a\)
ⓑ \(p{q}^{2}-4p-3{q}^{2}\)

Solution

ⓐ \(-10a+5b\)
ⓑ \(p{q}^{2}-4p-3{q}^{2}\)

Try it.

ⓐ \(\ \text{14x}-3y-13x\)
ⓑ \({a}^{2}b-4a-5a{b}^{2}\)

Try it.

ⓐ \(2{a}^{2}+{b}^{2}-6{a}^{2}\)
ⓑ \({x}^{2}y-3x+7x{y}^{2}\)

Solution

ⓐ \(-4{a}^{2}+{b}^{2}\)
ⓑ \({x}^{2}y-3x+7x{y}^{2}\)

Try it.

ⓐ \(5{u}^{2}+4{v}^{2}-6{u}^{2}\)
ⓑ \(\ \text{12a}+8b\)

Try it.

ⓐ \(x{y}^{2}-5x-5{y}^{2}\)
ⓑ \(\ \text{19y}+5z\)

Solution

ⓐ \(x{y}^{2}-5x-5{y}^{2}\)
ⓑ \(19y+5z\)

Try it.

\(\text{12}a+5b-22a+p{q}^{2}-4p-3{q}^{2}\)

Try it.

\(\text{14x}-3y-13x+{a}^{2}b-4a-5a{b}^{2}\)

Solution

\(x-3y+{a}^{2}b-4a-5a{b}^{2}\)

Try it.

\(2{a}^{2}+{b}^{2}-6{a}^{2}+{x}^{2}y-3x+7x{y}^{2}\)

Try it.

\(5{u}^{2}+4{v}^{2}-6{u}^{2}+\ \text{12a}+8b\)

Solution

\(\text{-}{u}^{2}+4{v}^{2}+\ \text{12a}+8b\)

Try it.

\(x{y}^{2}-5x-5{y}^{2}+\ \text{19y}+5z\)

Try it.

Add: \(4a,-3b,-8a\)

Solution

\(\text{-}\text{4a}-3b\)

Try it.

Add:\(\ \text{4x},3y,-3x\)

Try it.

Subtract \(5{x}^{6}\) from \(-12{x}^{6}\)

Solution

\(-17{x}^{6}\)

Try it.

Subtract \(2{p}^{4}\) from \(-7{p}^{4}\)

In the following exercises, add the polynomials.

Try it.

\((5{y}^{2}+12y+4)+(6{y}^{2}-8y+7)\)

Solution

\(11{y}^{2}+4y+11\)

Try it.

\((4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)\)

Try it.

\(({x}^{2}+6x+8)+(-4{x}^{2}+11x-9)\)

Solution

\(-3{x}^{2}+17x-1\)

Try it.

\(({y}^{2}+9y+4)+(-2{y}^{2}-5y-1)\)

Try it.

\((8{x}^{2}-5x+2)+(3{x}^{2}+3)\)

Solution

\(11{x}^{2}-5x+5\)

Try it.

\((7{x}^{2}-9x+2)+(6{x}^{2}-4)\)

Try it.

\((5{a}^{2}+8)+({a}^{2}-4a-9)\)

Solution

\(6{a}^{2}-4a-1\)

Try it.

\(({p}^{2}-6p-18)+(2{p}^{2}+11)\)

In the following exercises, subtract the polynomials.

Try it.

\((4{m}^{2}-6m-3)-(2{m}^{2}+m-7)\)

Solution

\(2{m}^{2}-7m+4\)

Try it.

\((3{b}^{2}-4b+1)-(5{b}^{2}-b-2)\)

Try it.

\(({a}^{2}+8a+5)-({a}^{2}-3a+2)\)

Solution

\(11a+3\)

Try it.

\(({b}^{2}-7b+5)-({b}^{2}-2b+9)\)

Try it.

\((12{s}^{2}-15s)-(s-9)\)

Solution

\(12{s}^{2}-16s+9\)

Try it.

\((10{r}^{2}-20r)-(r-8)\)

In the following exercises, subtract the polynomials.

Try it.

Subtract \((9{x}^{2}+2)\) from \((12{x}^{2}-x+6)\)

Solution

\(3{x}^{2}-x+4\)

Try it.

Subtract \((5{y}^{2}-y+12)\) from \((10{y}^{2}-8y-20)\)

Try it.

Subtract \((7{w}^{2}-4w+2)\) from \((8{w}^{2}-w+6)\)

Solution

\({w}^{2}+3w+4\)

Try it.

Subtract \((5{x}^{2}-x+12)\) from \((9{x}^{2}-6x-20)\)

In the following exercises, find the difference of the polynomials.

Try it.

Find the difference of \(({w}^{2}+w-42)\) and \(({w}^{2}-10w+24)\)

Solution

\(11w-66\)

Try it.

Find the difference of \(({z}^{2}-3z-18)\) and \(({z}^{2}+5z-20)\)

In the following exercises, add the polynomials.

Try it.

\((7{x}^{2}-2xy+6{y}^{2})+(3{x}^{2}-5xy)\)

Solution

\(10{x}^{2}-7xy+6{y}^{2}\)

Try it.

\((-5{x}^{2}-4xy-3{y}^{2})+(2{x}^{2}-7xy)\)

Try it.

\((7{m}^{2}+mn-8{n}^{2})+(3{m}^{2}+2mn)\)

Solution

\(10{m}^{2}+3mn-8{n}^{2}\)

Try it.

\((2{r}^{2}-3rs-2{s}^{2})+(5{r}^{2}-3rs)\)

In the following exercises, add or subtract the polynomials.

Try it.

\(({a}^{2}-{b}^{2})-({a}^{2}+3ab-4{b}^{2})\)

Solution

\(-3ab+3{b}^{2}\)

Try it.

\(({m}^{2}+2{n}^{2})-({m}^{2}-8mn-{n}^{2})\)

Try it.

\(({p}^{3}-3{p}^{2}q)+(2p{q}^{2}+4{q}^{3})-(3{p}^{2}q+p{q}^{2})\)

Solution

\({p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}\)

Try it.

\(({a}^{3}-2{a}^{2}b)+(a{b}^{2}+{b}^{3})-(3{a}^{2}b+4a{b}^{2})\)

Try it.

\(({x}^{3}-{x}^{2}y)-(4x{y}^{2}-{y}^{3})+(3{x}^{2}y-x{y}^{2})\)

Solution

\({x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}\)

Try it.

\(({x}^{3}-2{x}^{2}y)-(x{y}^{2}-3{y}^{3})-({x}^{2}y-4x{y}^{2})\)

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

Try it.

For the function \(f(x)=8{x}^{2}-3x+2,\) find:
ⓐ \(f(5)\) ⓑ \(f(-2)\) ⓒ \(f(0)\)

Solution

ⓐ 187 ⓑ 40 ⓒ 2

Try it.

For the function \(f(x)=5{x}^{2}-x-7,\) find:
ⓐ \(f(-4)\) ⓑ \(f(1)\) ⓒ \(f(0)\)

Try it.

For the function \(g(x)=4-36x,\) find:
ⓐ \(g(3)\) ⓑ \(g(0)\) ⓒ \(g(-1)\)

Solution

ⓐ \(-104\) ⓑ 4 ⓒ 40

Try it.

For the function \(g(x)=16-36{x}^{2},\) find:
ⓐ \(g(-1)\) ⓑ \(g(0)\) ⓒ \(g(2)\)

In the following exercises, find the height for each polynomial function.

Try it.

A painter drops a brush from a platform 75 feet high. The polynomial function \(h(t)=-16{t}^{2}+75\) gives the height of the brush t seconds after it was dropped. Find the height after \(t=2\) seconds.

Solution

The height is 11 feet.

Try it.

A girl drops a ball off a 200-foot cliff into the ocean. The polynomial \(h(t)=-16{t}^{2}+200\) gives the height of the ball, in feet, t seconds after it is dropped. Find the height after \(t=3\) seconds.

Try it.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial function \(R(p)=-4{p}^{2}+420p.\) Find the revenue received when \(p=60\) dollars.

Solution

The revenue is $10,800.

Try it.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial \(R(p)=-4{p}^{2}+420p.\) Find the revenue received when \(p=90\) dollars.

Try it.

The polynomial \(C(x)=6{x}^{2}+90x\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with \(x=4\) feet.

Solution

The cost is $456.

Try it.

The polynomial \(C(x)=6{x}^{2}+90x\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with \(x=6\) feet.

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

Try it.

\(f(x)=2{x}^{2}-4x+1\) and \(g(x)=5{x}^{2}+8x+3\)

Solution

ⓐ \((f+g)(x)=7{x}^{2}+4x+4\) ⓑ \((f+g)(2)=40\)
ⓒ \((f-g)(x)=-3{x}^{2}-12x-2\)
ⓓ \((f-g)(-3)=7\)

Try it.

\(f(x)=4{x}^{2}-7x+3\) and \(g(x)=4{x}^{2}+2x-1\)

Try it.

\(f(x)=3{x}^{3}-{x}^{2}-2x+3\) and \(g(x)=3{x}^{3}-7x\)

Solution

ⓐ  \((f+g)(x)=6{x}^{3}-{x}^{2}-9x+3\)
ⓑ \((f+g)(2)=29\)
ⓒ \((f-g)(x)=\text{-}{x}^{2}+5x+3\)
ⓓ \((f-g)(-3)=-21\)

Try it.

\(f(x)=5{x}^{3}-{x}^{2}+3x+4\) and \(g(x)=8{x}^{3}-1\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. The constant is called a coefficient. When it is of the form \(a{x}^{m}\), where \(a\) is a constant and \(m\) is a whole number, it is called a monomial. Some examples of monomial are \(8,-2{x}^{2},4{y}^{3},\ \text{and}\ 11{z}^{7}\).

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Here are some examples of polynomials.

\(\begin{array}{lllllllllllll}\text{Polynomial} & & & \ b+1 & & & \ 4{y}^{2}-7y+2 & & & \ 4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1 & & & \\ \text{Monomial} & & & \ 14 & & & \ 8{y}^{2} & & & \ -9{x}^{3}{y}^{5} & & & \ -13 \\ \text{Binomial} & & & \ a+7 & & & \ 4b-5 & & & \ {y}^{2}-16 & & & \ 3{x}^{3}-9{x}^{2} \\ \text{Trinomial} & & & \ {x}^{2}-7x+12 & & & \ 9{y}^{2}+2y-8 & & & \ 6{m}^{4}-{m}^{3}+8m & & & \ {z}^{4}+3{z}^{2}-1\end{array}\)

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

Example

Try it.

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

1. ⓐ \(4{y}^{2}-8y-6\)
2. ⓑ \(-5{a}^{4}{b}^{2}\)
3. ⓒ \(2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4\)
4. ⓓ \(13-5{m}^{3}\)
5. ⓔ \(q\)

Solution

	Polynomial	Number of terms	Type
ⓐ	\(4{y}^{2}-8y-6\)	\(3\)	Trinomial
ⓑ	\(-5{a}^{4}{b}^{2}\)	\(1\)	Monomial
ⓒ	\(2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4\)	\(5\)	Polynomial
ⓓ	\(13-5{m}^{3}\)	\(2\)	Binomial
ⓔ	\(q\)	\(1\)	Monomial

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0, it has no variable.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example

Try it.

Find the degree of the following polynomials.

1. ⓐ \(10y\)
2. ⓑ \(4{x}^{3}-7x+5\)
3. ⓒ \(-15\)
4. ⓓ \(-8{b}^{2}+9b-2\)
5. ⓔ \(8x{y}^{2}+2y\)

Solution

ⓐ The exponent of \(y\) is one. \(y={y}^{1}\)	\(10y\) The degree is 1.
ⓑ The highest degree of all the terms is 3.	\(4{x}^{3}-7x+5\) The degree is 3.
ⓒ The degree of a constant is 0.	\(-15\) The degree is 0.
ⓓ The highest degree of all the terms is 2.	\(-8{b}^{2}+9b-2\) The degree is 2.
ⓔ The highest degree of all the terms is 3.	\(8x{y}^{2}+2y\) The degree is 3.

Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Example

Try it.

Add: \(25{y}^{2}+15{y}^{2}\).

Solution

	\(25{y}^{2}+15{y}^{2}\)
Combine like terms.	\(40{y}^{2}\)

Example

Try it.

Subtract: \(16p-(-7p)\).

Solution

	\(16p-(-7p)\)
Combine like terms.	\(23p\)

Remember that like terms must have the same variables with the same exponents.

Example

Try it.

Simplify: \({c}^{2}+7{d}^{2}-6{c}^{2}\).

Solution

	\({c}^{2}+7{d}^{2}-6{c}^{2}\)
Combine like terms.	\(-5{c}^{2}+7{d}^{2}\)

Example

Try it.

Simplify: \({u}^{2}v+5{u}^{2}-3{v}^{2}\).

Solution

	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)
There are no like terms to combine.	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)

Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms, those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example

Try it.

Find the sum: \((5{y}^{2}-3y+15)+(3{y}^{2}-4y-11).\)

Solution

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.

Example

Try it.

Find the difference: \((9{w}^{2}-7w+5)-(2{w}^{2}-4).\)

Solution

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Example

Try it.

Subtract: \(({c}^{2}-4c+7)\) from \((7{c}^{2}-5c+3)\).

Solution

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Example

Try it.

Find the sum: \(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)\).

Solution

	\(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)\)
Distribute.	\({u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv\)
Rearrange the terms, to put like terms together.	\({u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}\)
Combine like terms.	\(4{u}^{2}-4uv+5{v}^{2}\)

Example

Try it.

Find the difference: \(({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})\).

Solution

	\(({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})\)
Distribute.	\({p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}\)
Rearrange the terms, to put like terms together.	\({p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}\)
Combine like terms.	\(-10pq+3{q}^{2}\)

Example

Try it.

Simplify: \(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2})\).

Solution

	\(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2})\)
Distribute.	\({a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}\)
Rearrange the terms, to put like terms together.	\({a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}\)
Combine like terms.	\({a}^{3}-{b}^{3}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

Example

Try it.

Evaluate \(5{x}^{2}-8x+4\) when

1. ⓐ \(x=4\)
2. ⓑ \(x=-2\)
3. ⓒ \(x=0\)

Solution

ⓐ \(x=4\)
Simplify the exponents.
Multiply.
Simplify.

ⓑ \(x=-2\)
Simplify the exponents.
Multiply.
Simplify.

ⓒ \(x=0\)
Simplify the exponents.
Multiply.
Simplify.

Example

Try it.

The polynomial \(-16{t}^{2}+250\) gives the height of a ball \(t\) seconds after it is dropped from a 250 foot tall building. Find the height after \(t=2\) seconds.

Solution

	\(-16{t}^{2}+250\)
Substitute \(t=2\).	\(-16{(2)}^{2}+250\)
Simplify.	\(-16\cdot 4+250\)
Simplify.	\(-64+250\)
Simplify.	\(186\)
	After 2 seconds the height of the ball is 186 feet.

Example

Try it.

The polynomial \(6{x}^{2}+15xy\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with \(x=4\) feet and \(y=6\) feet.

Solution

Simplify.
Simplify.
Simplify.
	The cost of producing the box is $456.

Key Concepts

• Monomials
  • A monomial is a term of the form \(a{x}^{m}\), where \(a\) is a constant and \(m\) is a whole number
• Polynomials
  • polynomial, A monomial, or two or more monomials combined by addition or subtraction is a polynomial.
  • monomial, A polynomial with exactly one term is called a monomial.
  • binomial, A polynomial with exactly two terms is called a binomial.
  • trinomial, A polynomial with exactly three terms is called a trinomial.
• Degree of a Polynomial
  • The degree of a term is the sum of the exponents of its variables.
  • The degree of a constant is 0.
  • The degree of a polynomial is the highest degree of all its terms.

Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Try it.

ⓐ \(81{b}^{5}-24{b}^{3}+1\) ⓑ \(5{c}^{3}+11{c}^{2}-c-8\) ⓒ \(\frac{14}{15}y+\frac{1}{7}\)
ⓓ 5
ⓔ \(4y+17\)

Solution

ⓐ trinomial ⓑ polynomial ⓒ binomial ⓓ monomial ⓔ binomial

Try it.

ⓐ \({x}^{2}-{y}^{2}\) ⓑ \(-13{c}^{4}\) ⓒ \({x}^{2}+5x-7\) ⓓ \({x}^{2}{y}^{2}-2xy+8\) ⓔ 19

Try it.

ⓐ \(8-3x\) ⓑ \({z}^{2}-5z-6\) ⓒ \({y}^{3}-8{y}^{2}+2y-16\) ⓓ \(81{b}^{5}-24{b}^{3}+1\) ⓔ \(-18\)

Solution

ⓐ binomial ⓑ trinomial ⓒ polynomial ⓓ trinomial ⓔ monomial

Try it.

ⓐ \(11{y}^{2}\) ⓑ \(-73\) ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\) ⓓ \(4y+17\) ⓔ \(5{c}^{3}+11{c}^{2}-c-8\)

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Try it.

ⓐ \(6{a}^{2}+12a+14\) ⓑ \(18x{y}^{2}z\) ⓒ \(5x+2\) ⓓ \({y}^{3}-8{y}^{2}+2y-16\) ⓔ \(-24\)

Solution

ⓐ 2 ⓑ 4 ⓒ 1 ⓓ 3 ⓔ 0

Try it.

ⓐ \(9{y}^{3}-10{y}^{2}+2y-6\) ⓑ \(-12{p}^{4}\) ⓒ \({a}^{2}+9a+18\) ⓓ \(20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30\) ⓔ 17

Try it.

ⓐ \(14-29x\) ⓑ \({z}^{2}-5z-6\) ⓒ \({y}^{3}-8{y}^{2}+2y-16\) ⓓ \(23a{b}^{2}-14\) ⓔ \(-3\)

Solution

ⓐ 1 ⓑ 2 ⓒ 3 ⓓ 3 ⓔ 0

Try it.

ⓐ \(62{y}^{2}\) ⓑ 15 ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\) ⓓ \(10-9x\) ⓔ \({m}^{4}+4{m}^{3}+6{m}^{2}+4m+1\)

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Try it.

\({\ \text{7x}}^{2}+5{x}^{2}\)

Solution

\(12{x}^{2}\)

Try it.

\({\ \text{4y}}^{3}+6{y}^{3}\)

Try it.

\(-12w+18w\)

Solution

\(6w\)

Try it.

\(\text{4a}-9a\)

Solution

\(-5a\)

Try it.

\(\text{-}y-5y\)

Try it.

\(28x-(-12x)\)

Solution

\(40x\)

Try it.

\(13z-(-4z)\)

Try it.

\(-10x-35x\)

Try it.

\(12a+5b-22a\)

Solution

\(\text{-10a}+5b\)

Try it.

\(\text{14x}-3y-13x\)

Try it.

\(2{a}^{2}+{b}^{2}-6{a}^{2}\)

Solution

\(-4{a}^{2}+{b}^{2}\)

Try it.

\(5{u}^{2}+4{v}^{2}-6{u}^{2}\)

Try it.

\(x{y}^{2}-5x-5{y}^{2}\)

Solution

\(x{y}^{2}-5x-5{y}^{2}\)

Try it.

\(p{q}^{2}-4p-3{q}^{2}\)

Try it.

\({a}^{2}b-4a-5a{b}^{2}\)

Solution

\({a}^{2}b-4a-5a{b}^{2}\)

Try it.

\({x}^{2}y-3x+7x{y}^{2}\)

Try it.

\(\text{12a}+8b\)

Solution

\(\text{12a}+8b\)

Try it.

\(\text{19y}+5z\)

Try it.

Add: \(4a,-3b,-8a\)

Solution

\(-4a-3b\)

Try it.

Add: \(\ \text{4x},3y,-3x\)

Try it.

Subtract \(5{x}^{6}\text{from}-12{x}^{6}\).

Solution

\(-17{x}^{6}\)

Try it.

Subtract \(2{p}^{4}\text{from}-7{p}^{4}\).

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Try it.

\((5{y}^{2}+12y+4)+(6{y}^{2}-8y+7)\)

Solution

\(11{y}^{2}+4y+11\)

Try it.

\((4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)\)

Try it.

\(({x}^{2}+6x+8)+(-4{x}^{2}+11x-9)\)

Solution

\(-3{x}^{2}+17x-1\)

Try it.

\(({y}^{2}+9y+4)+(-2{y}^{2}-5y-1)\)

Try it.

\((8{x}^{2}-5x+2)+(3{x}^{2}+3)\)

Solution

\(11{x}^{2}-5x+5\)

Try it.

\((7{x}^{2}-9x+2)+(6{x}^{2}-4)\)

Try it.

\((5{a}^{2}+8)+({a}^{2}-4a-9)\)

Solution

\(6{a}^{2}-4a-1\)

Try it.

\(({p}^{2}-6p-18)+(2{p}^{2}+11)\)

Try it.

\((4{m}^{2}-6m-3)-(2{m}^{2}+m-7)\)

Solution

\(2{m}^{2}-7m+4\)

Try it.

\((3{b}^{2}-4b+1)-(5{b}^{2}-b-2)\)

Try it.

\(({a}^{2}+8a+5)-({a}^{2}-3a+2)\)

Solution

\(11a+3\)

Try it.

\(({b}^{2}-7b+5)-({b}^{2}-2b+9)\)

Try it.

\((12{s}^{2}-15s)-(s-9)\)

Solution

\(12{s}^{2}-16s+9\)

Try it.

\((10{r}^{2}-20r)-(r-8)\)

Try it.

Subtract \((9{x}^{2}+2)\) from \((12{x}^{2}-x+6)\).

Solution

\(3{x}^{2}-x+4\)

Try it.

Subtract \((5{y}^{2}-y+12)\) from \((10{y}^{2}-8y-20)\).

Try it.

Subtract \((7{w}^{2}-4w+2)\) from \((8{w}^{2}-w+6)\).

Solution

\({w}^{2}+3w+4\)

Try it.

Subtract \((5{x}^{2}-x+12)\) from \((9{x}^{2}-6x-20)\).

Try it.

Find the sum of \((2{p}^{3}-8)\) and \(({p}^{2}+9p+18)\).

Solution

\(2{p}^{3}+{p}^{2}+9p+10\)

Try it.

Find the sum of
\(({q}^{2}+4q+13)\) and \((7{q}^{3}-3)\).

Try it.

Find the sum of \((8{a}^{3}-8a)\) and \(({a}^{2}+6a+12)\).

Solution

\(8{a}^{3}+{a}^{2}-2a+12\)

Try it.

Find the sum of
\(({b}^{2}+5b+13)\) and \((4{b}^{3}-6)\).

Try it.

Find the difference of
\(({w}^{2}+w-42)\) and
\(({w}^{2}-10w+24)\).

Solution

\(11w-66\)

Try it.

Find the difference of
\(({z}^{2}-3z-18)\) and
\(({z}^{2}+5z-20)\).

Try it.

Find the difference of
\(({c}^{2}+4c-33)\) and
\(({c}^{2}-8c+12)\).

Solution

\(12c-45\)

Try it.

Find the difference of
\(({t}^{2}-5t-15)\) and
\(({t}^{2}+4t-17)\).

Try it.

\((7{x}^{2}-2xy+6{y}^{2})+(3{x}^{2}-5xy)\)

Solution

\(10{x}^{2}-7xy+6{y}^{2}\)

Try it.

\((-5{x}^{2}-4xy-3{y}^{2})+(2{x}^{2}-7xy)\)

Try it.

\((7{m}^{2}+mn-8{n}^{2})+(3{m}^{2}+2mn)\)

Solution

\(10{m}^{2}+3mn-8{n}^{2}\)

Try it.

\((2{r}^{2}-3rs-2{s}^{2})+(5{r}^{2}-3rs)\)

Try it.

\(({a}^{2}-{b}^{2})-({a}^{2}+3ab-4{b}^{2})\)

Solution

\(-3ab+3{b}^{2}\)

Try it.

\(({m}^{2}+2{n}^{2})-({m}^{2}-8mn-{n}^{2})\)

Try it.

\(({u}^{2}-{v}^{2})-({u}^{2}-4uv-3{v}^{2})\)

Solution

\(4uv+2{v}^{2}\)

Try it.

\(({j}^{2}-{k}^{2})-({j}^{2}-8jk-5{k}^{2})\)

Try it.

\(({p}^{3}-3{p}^{2}q)+(2p{q}^{2}+4{q}^{3})-(3{p}^{2}q+p{q}^{2})\)

Solution

\({p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}\)

Try it.

\(({a}^{3}-2{a}^{2}b)+(a{b}^{2}+{b}^{3})-(3{a}^{2}b+4a{b}^{2})\)

Try it.

\(({x}^{3}-{x}^{2}y)-(4x{y}^{2}-{y}^{3})+(3{x}^{2}y-x{y}^{2})\)

Solution

\({x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}\)

Try it.

\(({x}^{3}-2{x}^{2}y)-(x{y}^{2}-3{y}^{3})-({x}^{2}y-4x{y}^{2})\)

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Try it.

Evaluate \(8{y}^{2}-3y+2\) when:

ⓐ \(y=5\) ⓑ \(y=-2\) ⓒ \(y=0\)

Solution

ⓐ 187 ⓑ 40 ⓒ 2

Try it.

Evaluate \(5{y}^{2}-y-7\) when:

ⓐ \(y=-4\) ⓑ \(y=1\) ⓒ \(y=0\)

Try it.

Evaluate \(4-36x\) when:

ⓐ \(x=3\) ⓑ \(x=0\) ⓒ \(x=-1\)

Solution

ⓐ −104 ⓑ 4 ⓒ 40

Try it.

Evaluate \(16-36{x}^{2}\) when:

ⓐ \(x=-1\) ⓑ \(x=0\) ⓒ \(x=2\)

Try it.

A painter drops a brush from a platform 75 feet high. The polynomial \(-16{t}^{2}+75\) gives the height of the brush \(t\) seconds after it was dropped. Find the height after \(t=2\) seconds.

Solution

11

Try it.

A girl drops a ball off a cliff into the ocean. The polynomial \(-16{t}^{2}+250\) gives the height of a ball \(t\) seconds after it is dropped from a 250-foot tall cliff. Find the height after \(t=2\) seconds.

Try it.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial \(-4{p}^{2}+420p.\) Find the revenue received when \(p=60\) dollars.

Solution

$10,800

Try it.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial \(-4{p}^{2}+420p.\) Find the revenue received when \(p=90\) dollars.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Identify Polynomials, Monomials, Binomials, and Trinomials

In Evaluate, Simplify, and Translate Expressions, you learned that a term is a constant or the product of a constant and one or more variables. The constant is called a coefficient. When it is of the form \(a{x}^{m},\) where \(a\) is a constant and \(m\) is a whole number, it is called a monomial. A monomial, or a sum and/or difference of monomials, is called a polynomial.

Notice the roots:

• poly- means many
• mono- means one
• bi- means two
• tri- means three

Here are some examples of polynomials:

Polynomial	\(b+1\)	\(4{y}^{2}-7y+2\)	\(5{x}^{5}-4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1\)
Monomial	\(5\)	\(4{b}^{2}\)	\(-9{x}^{3}\)
Binomial	\(3a-7\)	\({y}^{2}-9\)	\(17{x}^{3}+14{x}^{2}\)
Trinomial	\({x}^{2}-5x+6\)	\(4{y}^{2}-7y+2\)	\(5{a}^{4}-3{a}^{3}+a\)

Notice that every monomial, binomial, and trinomial is also a polynomial. They are special members of the family of polynomials and so they have special names. We use the words ‘monomial’, ‘binomial’, and ‘trinomial’ when referring to these special polynomials and just call all the rest ‘polynomials’.

Example

Try it.

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

1. ⓐ \(\ 8{x}^{2}-7x-9\)
2. ⓑ \(\ -5{a}^{4}\)
3. ⓒ \(\ {x}^{4}-7{x}^{3}-6{x}^{2}+5x+2\)
4. ⓓ \(\ 11-4{y}^{3}\)
5. ⓔ \(\ n\)

Solution

	Polynomial	Number of terms	Type
ⓐ	\(8{x}^{2}-7x-9\)	3	Trinomial
ⓑ	\(-5{a}^{4}\)	1	Monomial
ⓒ	\({x}^{4}-7{x}^{3}-6{x}^{2}+5x+2\)	5	Polynomial
ⓓ	\(11-4{y}^{3}\)	2	Binomial
ⓔ	\(n\)	1	Monomial

Determine the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is \(0\), it has no variable.

Let's see how this works by looking at several polynomials. We'll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Remember: Any base written without an exponent has an implied exponent of \(1.\)

Example

Try it.

Find the degree of the following polynomials:

1. ⓐ \(\ 4x\)
2. ⓑ \(\ 3{x}^{3}-5x+7\)
3. ⓒ \(\ -11\)
4. ⓓ \(\ -6{x}^{2}+9x-3\)
5. ⓔ \(\ 8x+2\)

Solution

ⓐ	\(4x\)
The exponent of \(x\) is one. \(x={x}^{1}\)	The degree is 1.
ⓑ	\(3{x}^{3}-5x+7\)
The highest degree of all the terms is 3.	The degree is 3
ⓒ	\(-11\)
The degree of a constant is 0.	The degree is 0.
ⓓ	\(-6{x}^{2}+9x-3\)
The highest degree of all the terms is 2.	The degree is 2.
ⓔ	\(8x+2\)
The highest degree of all the terms is 1.	The degree is 1.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Look back at the polynomials in . Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.

Add and Subtract Monomials

In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

Example

Try it.

Add: \(17{x}^{2}+6{x}^{2}.\)

Solution

	\(17{x}^{2}+6{x}^{2}\)
Combine like terms.	\(23{x}^{2}\)

Example

Try it.

Subtract: \(11n-(-8n).\)

Solution

	\(11n-(-8n)\)
Combine like terms.	\(19n\)

Example

Try it.

Simplify: \({a}^{2}+4{b}^{2}-7{a}^{2}.\)

Solution

	\({a}^{2}+4{b}^{2}-7{a}^{2}\)
Combine like terms.	\(-6{a}^{{}^{2}}+4{b}^{2}\)

Remember, \(-6{a}^{2}\) and \(4{b}^{2}\) are not like terms. The variables are not the same.

Add and Subtract Polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms, those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

Example

Try it.

Find the sum: \((4{x}^{2}-5x+1)+(3{x}^{2}-8x-9).\)

Solution

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.

Parentheses are grouping symbols. When we add polynomials as we did in , we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

Example

Try it.

Find the difference: \((7{u}^{2}-5u+3)-(4{u}^{2}-2).\)

Solution

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Example

Try it.

Subtract: \(({m}^{2}-3m+8)\) from \((9{m}^{2}-7m+4).\)

Solution

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Evaluate a Polynomial for a Given Value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials, substitute the given value for the variable into the polynomial, and then simplify.

Example

Try it.

Evaluate \(3{x}^{2}-9x+7\) when

1. ⓐ \(\ x=3\)
2. ⓑ \(\ x=-1\)

Solution

ⓐ \(x=3\)
	\(3{x}^{2}-9x+7\)
Substitute 3 for \(x\)	\(3{(3)}^{2}-9(3)+7\)
Simplify the expression with the exponent.	\(3\cdot 9-9(3)+7\)
Multiply.	\(27-27+7\)
Simplify.	\(7\)

ⓑ \(x=-1\)
	\(3{x}^{2}-9x+7\)
Substitute −1 for \(x\)	\(3{(-1)}^{2}-9(-1)+7\)
Simplify the expression with the exponent.	\(3\cdot 1-9(-1)+7\)
Multiply.	\(3+9+7\)
Simplify.	\(19\)

Example

Try it.

The polynomial \(-16{t}^{2}+300\) gives the height of an object \(t\) seconds after it is dropped from a \(300\) foot tall bridge. Find the height after \(t=3\) seconds.

Solution

Substitute 3 for \(t\)
Simplify the expression with the exponent.
Multiply.
Simplify.

Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.

Try it.

\({z}^{2}-5z-6\)

Try it.

\({a}^{2}+9a+18\)

Solution

trinomial

Try it.

\(-12{p}^{4}\)

Try it.

\({y}^{3}-8{y}^{2}+2y-16\)

Solution

polynomial

Try it.

\(23{y}^{2}\)

Solution

monomial

Try it.

\({m}^{4}+4{m}^{3}+6{m}^{2}+4m+1\)

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Try it.

\(8{a}^{5}-2{a}^{3}+1\)

Solution

5

Try it.

\(5{c}^{3}+11{c}^{2}-c-8\)

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Try it.

\({\text{6x}}^{2}+\ 9{x}^{2}\)

Solution

15x²

Try it.

\({\text{4y}}^{3}+\ 6{y}^{3}\)

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\(-12u\ +\ 4u\)

Solution

−8u

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\(-3m\ +\ 9m\)

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\(5a\ +\ 7b\)

Solution

5a + 7b

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\(8y\ +\ 6z\)

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Add: \(\ 4a\ ,\ -3b,\ -8a\)

Solution

−4a −3b

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Add: \(4x\ ,\ 3y\ ,\ -3x\)

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Subtract \(5{x}^{6}\ \text{from}\ -12{x}^{6}\)

Solution

−17x⁶

Try it.

Subtract \(2{p}^{4}\ \text{from}\ -7{p}^{4}\)

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Try it.

\((4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)\)

Solution

12y² + 4y + 8

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\((7{x}^{2}-9x+2)+(6{x}^{2}-4x+3)\)

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\(({x}^{2}+6x+8)+(-4{x}^{2}+11x-9)\)

Solution

−3x² + 17x − 1

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\(({y}^{2}+9y+4)+(-2{y}^{2}-5y-1)\)

Try it.

\((3{a}^{2}+7)+({a}^{2}-7a-18)\)

Solution

4a² − 7a − 11

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\(({p}^{2}-5p-11)+(3{p}^{2}+9)\)

Try it.

\((6{m}^{2}-9m-3)-(2{m}^{2}+m-5)\)

Solution

4m² − 10m + 2

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\((3{n}^{2}-4n+1)-(4{n}^{2}-n-2)\)

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\(({z}^{2}+8z+9)-({z}^{2}-3z+1)\)

Solution

11z + 8

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\(({z}^{2}-7z+5)-({z}^{2}-8z+6)\)

Try it.

\((12{s}^{2}-15s)-(s-9)\)

Solution

12s² − 16s + 9

Try it.

\((10{r}^{2}-20r)-(r-8)\)

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Find the sum of \((2{p}^{3}-8)\) and \(({p}^{2}+9p+18)\)

Solution

2p³ + p² + 9p + 10

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Find the sum of \(({q}^{2}+4q+13)\) and \((7{q}^{3}-3)\)

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Subtract \((7{x}^{2}-4x+2)\) from \((8{x}^{2}-x+6)\)

Solution

x² + 3x + 4

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Subtract \((5{x}^{2}-x+12)\) from \((9{x}^{2}-6x-20)\)

Try it.

Find the difference of \(({w}^{2}+w-42)\) and \(({w}^{2}-10w+24)\)

Solution

11w − 66

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Find the difference of \(({z}^{2}-3z-18)\) and \(({z}^{2}+5z-20)\)

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

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\(\text{Evaluate}\ 8{y}^{2}-3y+2\)

1. ⓐ \(\ y=5\)
2. ⓑ \(\ y=-2\)
3. ⓒ \(\ y=0\)

Solution

1. ⓐ 187
2. ⓑ 40
3. ⓒ 2

Try it.

\(\text{Evaluate}\ 5{y}^{2}-y-7\ \text{when:}\)

1. ⓐ \(\ y=-4\)
2. ⓑ \(\ y=1\)
3. ⓒ \(y=0\)

Try it.

\(\text{Evaluate}\ 4-36x\ \text{when:}\)

1. ⓐ \(\ x=3\)
2. ⓑ \(\ x=0\)
3. ⓒ \(x=-1\)

Solution

1. ⓐ −104
2. ⓑ 4
3. ⓒ 40

Try it.

\(\text{Evaluate}\ 16-36{x}^{2}\ \text{when:}\)

1. ⓐ \(\ x=-1\)
2. ⓑ \(\ x=0\)
3. ⓒ \(x=2\)

Try it.

A window washer drops a squeegee from a platform \(275\) feet high. The polynomial \(-16{t}^{2}+275\) gives the height of the squeegee \(t\) seconds after it was dropped. Find the height after \(t=4\) seconds.

Solution

19 feet

Try it.

A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p dollars each is given by the polynomial \(-5{p}^{2}+350p.\) Find the revenue received when \(p=50\) dollars.

Condensed: the full section is in OpenStax Prealgebra 2e.

Now you Ոչ մի հաշվիչ չի կարող լուծել այս խնդիրը, բայց դրա մասերը հաշվարկելի են։ Փորձեք ստորեւ նշվածներից մեկը կամ գրեք ձեր սեփականը։

⌨ Տեսակ ✎ Նկարել

Ձեր մաթեմատիկական խնդիրը

π Լրացնել

Պարզաբանում (40)

Սկզբում փորձեք յուրաքանչյուրը թղթի վրա։ Առաջին հերթին բացահայտեք պատասխանը ստուգելու համար։ Ստուգված պատասխանները կարող են բացվել յուրաքանչյուր քայլում։

1. Simplify: \(3{x}^{2}+3x+1+8{x}^{2}+5x+5.\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   \(11{x}^{2}+8x+6\)

2. Subtract: \((5n+8)-(2n-1).\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   \(3n+9\)

3. Evaluate: \(4x{y}^{2}\) when \(x=-2\) and \(y=5.\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   \(-200\)

4. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

   ⓐ \(7{y}^{2}-5y+3\) ⓑ \(-2{a}^{4}{b}^{2}\) ⓒ \(3{x}^{5}-4{x}^{3}-6{x}^{2}+x-8\) ⓓ \(2y-8x{y}^{3}\) ⓔ 15

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   	Polynomial	Number of terms	Type	Degree of terms	Degree of polynomial
   ⓐ	\(7{y}^{2}-5y+3\)	3	Trinomial	2, 1, 0	2
   ⓑ	\(-2{a}^{4}{b}^{2}\)	1	Monomial	6	6
   ⓒ	\(3{x}^{5}-4{x}^{3}-6{x}^{2}+x-8\)	5	Polynomial	5, 3, 2, 1, 0	5
   ⓓ	\(2y-8x{y}^{3}\)	2	Binomial	1, 4	4
   ⓔ	15	1	Monomial	0	0

5. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

   ⓐ \(-5\) ⓑ \(8{y}^{3}-7{y}^{2}-y-3\) ⓒ \(-3{x}^{2}y-5xy+9x{y}^{3}\) ⓓ \(81{m}^{2}-4{n}^{2}\) ⓔ \(-3{x}^{6}{y}^{3}z\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   ⓐ monomial, 0
   ⓑ polynomial, 3 ⓒ trinomial, 4
   ⓓ binomial, 2 ⓔ monomial, 10

6. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

   ⓐ \(64{k}^{3}-8\) ⓑ \(9{m}^{3}+4{m}^{2}-2\) ⓒ \(\frac{5}{6}\) ⓓ \(8{a}^{4}-7{a}^{3}b-6{a}^{2}{b}^{2}-4a{b}^{3}+7{b}^{4}\) ⓔ \(\text{-}{p}^{4}{q}^{3}\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   ⓐ binomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7

7. Add or subtract: ⓐ \(25{y}^{2}+15{y}^{2}\) ⓑ \(16p{q}^{3}-(-7p{q}^{3}).\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   ⓐ
   \(\begin{array}{llll} & & & \ 25{y}^{2}+15{y}^{2} \\ \text{Combine like terms.} & & & \ 40{y}^{2}\end{array}\)

   ⓑ
   \(\begin{array}{llll} & & & \ 16p{q}^{3}-(-7p{q}^{3}) \\ \text{Combine like terms.} & & & \ 23p{q}^{3}\end{array}\)

8. Add or subtract: ⓐ \(12{q}^{2}+9{q}^{2}\) ⓑ \(8m{n}^{3}-(-5m{n}^{3}).\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   ⓐ \(21{q}^{2}\) ⓑ \(13m{n}^{3}\)

9. Add or subtract: ⓐ \(-15{c}^{2}+8{c}^{2}\) ⓑ \(-15{y}^{2}{z}^{3}-(-5{y}^{2}{z}^{3}).\)

   Աշխատել դրա վրա Խորհուրդ

   Առաջարկել պատասխանը

   ⓐ \(-7{c}^{2}\) ⓑ \(-10{y}^{2}{z}^{3}\)

10. Simplify: ⓐ \({a}^{2}+7{b}^{2}-6{a}^{2}\) ⓑ \({u}^{2}v+5{u}^{2}-3{v}^{2}.\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ

    	\(\ {a}^{2}+7{b}^{2}-6{a}^{2}\)
    Combine like terms.	\(\ -5{a}^{2}+7{b}^{2}\)

    ⓑ

    	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)
    There are no like terms to combine. In this case, the polynomial is unchanged.	\({u}^{2}v+5{u}^{2}-3{v}^{2}\)

11. Add: ⓐ \(8{y}^{2}+3{z}^{2}-3{y}^{2}\) ⓑ \({m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ \(5{y}^{2}+3{z}^{2}\)
    ⓑ \({m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}\)

12. Add: ⓐ \(3{m}^{2}+{n}^{2}-7{m}^{2}\) ⓑ \(p{q}^{2}-6p-5{q}^{2}.\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ \(-4{m}^{2}+{n}^{2}\)
    ⓑ \(p{q}^{2}-6p-5{q}^{2}\)

13. Find the sum:\((7{y}^{2}-2y+9)+(4{y}^{2}-8y-7).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    Identify like terms.	\((\underline{\underline{7{y}^{2}}}-\underline{2y}+9)+(\underline{\underline{4{y}^{2}}}-\underline{8y}-7)\)
    Rewrite without the parentheses, rearranging to get the like terms together.	\(\underline{\underline{7{y}^{2}+4{y}^{2}}}-\underline{2y-8y}+9-7\)
    Combine like terms.	\(11{y}^{2}-10y+2\)

14. Find the sum: \((7{x}^{2}-4x+5)+({x}^{2}-7x+3).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(8{x}^{2}-11x+8\)

15. Find the sum: \((14{y}^{2}+6y-4)+(3{y}^{2}+8y+5).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(17{y}^{2}+14y+1\)

16. Find the difference: \((9{w}^{2}-7w+5)-(2{w}^{2}-4).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    	\((9{w}^{2}-7w+5)-(2{w}^{2}-4)\)
    Distribute and identify like terms.	\(\underline{\underline{9{w}^{2}}}-\underline{7w}+5-\underline{\underline{2{w}^{2}}}+4\)
    Rearrange the terms.	\(\underline{\underline{9{w}^{2}-2{w}^{2}}}-\underline{7w}+5+4\)
    Combine like terms.	\(7{w}^{2}-7w+9\)

17. Find the difference: \((8{x}^{2}+3x-19)-(7{x}^{2}-14).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \({x}^{2}+3x-5\)

18. Find the difference: \((9{b}^{2}-5b-4)-(3{b}^{2}-5b-7).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(6{b}^{2}+3\)

19. Subtract \(({p}^{2}+10pq-2{q}^{2})\) from \(({p}^{2}+{q}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    	\(({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})\)
    Distribute.	\({p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}\)
    Rearrange the terms, to put like terms together.	\({p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}\)
    Combine like terms.	\(-10pq+3{q}^{2}\)

20. Subtract \(({a}^{2}+5ab-6{b}^{2})\) from \(({a}^{2}+{b}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(-5ab+7{b}^{2}\)

21. Subtract \(({m}^{2}-7mn-3{n}^{2})\) from \(({m}^{2}+{n}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(7mn+4{n}^{2}\)

22. Find the sum: \(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    	\(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)\)
    Distribute.	\({u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv\)
    Rearrange the terms to put like terms together.	\({u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}\)
    Combine like terms.	\(4{u}^{2}-4uv+5{v}^{2}\)

23. Find the sum: \((3{x}^{2}-4xy+5{y}^{2})+(2{x}^{2}-xy).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(5{x}^{2}-5xy+5{y}^{2}\)

24. Find the sum: \((2{x}^{2}-3xy-2{y}^{2})+(5{x}^{2}-3xy).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \(7{x}^{2}-6xy-2{y}^{2}\)

25. Simplify: \(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    	\(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2})\)
    Distribute.	\({a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}\)
    Rewrite without the parentheses, rearranging to get the like terms together.	\({a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}\)
    Combine like terms.	\({a}^{3}-{b}^{3}\)

26. Simplify: \(({x}^{3}-{x}^{2}y)-(x{y}^{2}+{y}^{3})+({x}^{2}y+x{y}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \({x}^{3}-{y}^{3}\)

27. Simplify: \(({p}^{3}-{p}^{2}q)+(p{q}^{2}+{q}^{3})-({p}^{2}q+p{q}^{2}).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    \({p}^{3}-2{p}^{2}q+{q}^{3}\)

28. For the function \(f(x)=5{x}^{2}-8x+4\) find: ⓐ \(f(4)\) ⓑ \(f(-2)\) ⓒ \(f(0).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ
    

    Simplify the exponents.
    Multiply.
    Simplify.

    ⓑ
    

    Simplify the exponents.
    Multiply.
    Simplify.

    ⓒ
    

    Simplify the exponents.
    Multiply.
    Simplify.

29. For the function \(f(x)=3{x}^{2}+2x-15,\) find ⓐ \(f(3)\) ⓑ \(f(-5)\) ⓒ \(f(0).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ 18 ⓑ 50 ⓒ \(-15\)

30. For the function \(g(x)=5{x}^{2}-x-4,\) find ⓐ \(g(-2)\) ⓑ \(g(-1)\) ⓒ \(g(0).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ 18 ⓑ 2 ⓒ \(-4\)

31. The polynomial function \(h(t)=-16{t}^{2}+250\) gives the height of a ball t seconds after it is dropped from a 250-foot tall building. Find the height after \(t=2\) seconds.

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    	\(h(t)=-16{t}^{2}+250\)
    To find \(h(2),\) substitute \(t=2.\)	\(h(2)=-16{(2)}^{2}+250\)
    Simplify.	\(h(2)=-16\cdot 4+250\)
    Simplify.	\(h(2)=-64+250\)
    Simplify.	\(h(2)=186\)
    	After 2 seconds the height of the ball is 186 feet.

32. The polynomial function \(h(t)=-16{t}^{2}+150\) gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after \(t=0\) seconds (the initial height of the object).

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    The height is \(150\) feet.

33. The polynomial function \(h(t)=-16{t}^{2}+175\) gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after \(t=3\) seconds.

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    The height is 31 feet.

34. For functions \(f(x)=3{x}^{2}-5x+7\) and \(g(x)={x}^{2}-4x-3,\) find:

    ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ
    

    Rewrite without the parentheses.
    Put like terms together.
    Combine like terms.

    ⓑ In part (a) we found \((f+g)(x)\) and now are asked to find \((f+g)(3).\)
    

    	\((f+g)(x)=4{x}^{2}-9x+4\)
    To find \((f+g)(3),\) substitute \(x=3.\)	\((f+g)(3)=4{(3)}^{2}-9\cdot 3+4\)
    	\((f+g)(3)=4\cdot 9-9\cdot 3+4\)
    	\((f+g)(3)=36-27+4\)

    Notice that we could have found \((f+g)(3)\) by first finding the values of \(f(3)\) and \(g(3)\) separately and then adding the results.

    Find \(f(3).\)
    Find \(g(3).\)
    Find \((f+g)(3).\)
    \(\)

    ⓒ
    

    Rewrite without the parentheses.
    Put like terms together.
    Combine like terms.

    ⓓ
    

35. For functions \(f(x)=2{x}^{2}-4x+3\) and \(g(x)={x}^{2}-2x-6,\) find: ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ \((f+g)(x)=3{x}^{2}-6x-3\) ⓑ \((f+g)(3)=6\)
    ⓒ \((f-g)(x)={x}^{2}-2x+9\)
    ⓓ \((f-g)(-2)=17\)

36. For functions \(f(x)=5{x}^{2}-4x-1\) and \(g(x)={x}^{2}+3x+8,\) find ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ \((f+g)(x)=6{x}^{2}-x+7\) ⓑ \((f+g)(3)=58\)
    ⓒ \((f-g)(x)=4{x}^{2}-7x-9\)
    ⓓ \((f-g)(-2)=21\)

37. ⓐ \(47{x}^{5}-17{x}^{2}{y}^{3}+{y}^{2}\)
    ⓑ \(5{c}^{3}+11{c}^{2}-c-8\)
    ⓒ \(\frac{5}{9}ab+\frac{1}{3}b\)
    ⓓ 4
    ⓔ \(4pq+17\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
    ⓔ binomial, 2

38. ⓐ \({x}^{2}-{y}^{2}\)
    ⓑ \(-13{c}^{4}\)
    ⓒ \({a}^{2}+2ab-7{b}^{2}\)
    ⓓ \(4{x}^{2}{y}^{2}-3xy+8\)
    ⓔ 19

    Աշխատել դրա վրա Խորհուրդ

39. ⓐ \(8y-5x\)
    ⓑ \({y}^{2}-5yz-6{z}^{2}\)
    ⓒ \({y}^{3}-8{y}^{2}+2y-16\)
    ⓓ \(81a{b}^{4}-24{a}^{2}{b}^{2}+3b\)
    ⓔ \(-18\)

    Աշխատել դրա վրա Խորհուրդ

    Առաջարկել պատասխանը

    ⓐ binomial, 1ⓑ trinomial, 2
    ⓒ polynomial, 3ⓓ trinomial, 5
    ⓔ monomial, 0

40. ⓐ \(11{y}^{2}\)
    ⓑ \(-73\)
    ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\)
    ⓓ \(4{y}^{2}+17{z}^{2}\)
    ⓔ \(5{c}^{3}+11{c}^{2}-c-8\)

    Աշխատել դրա վրա Խորհուրդ

Ձեր սեփական աշխատանքը շարունակեք

Ազատ հաշիվը յուրաքանչյուր դասավանդման համար գրառումներ է ավելացնում, Ձեր ավարտելիք դասերի և լուծված խնդիրների ցուցակ է պահում մեկ տեղում, և ուսուցիչ, ում կարող եք հարցնել այս էջի մասին։ Մաթեմատիկան բաց է բոլորի համար, անկախ նրանից մուտքագրվել եք արդյոք կամ ոչ։

Գրանցվել Գրանցում

Օգտագործված սիմվոլներ

Տեղադրեք ցանկացած նշան՝ ամբողջական սահմանման, նկարի և դրա մեջ ամեն մի տառի իմաստը տեսնելու համար։

\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Ինչպես: Add and Subtract Polynomials

1. Identify polynomials, monomials, binomials, and trinomials
2. Determine the degree of polynomials
3. Add and subtract monomials
4. Add and subtract polynomials
5. Evaluate a polynomial for a given value

Հարցեր, որոնք մարդիկ տալիս են :

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Այս էջը կառուցված է հետևյալ սկզբունքով. OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Ընդհանուր և հատուկ դասակարգումներ: Ընդհանուր դասակարգում. Քանի որ դրա աղբյուրը թույլատրված է ոչ առևտրային օգտագործման համար, այս դասընթացը անվճար է, ոչինչ չի վաճառվում և կիսվում է նույն թույլտվության ներքո։

Ցուցադրել Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value