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.	\((\underset{____}{\underset{____}{7{y}^{2}}}-\underset{___}{2y}+9)+(\underset{____}{\underset{____}{4{y}^{2}}}-\underset{___}{8y}-7)\)
Rewrite without the parentheses, rearranging to get the like terms together.	\(\underset{_________}{\underset{_________}{7{y}^{2}+4{y}^{2}}}-\underset{_______}{2y-8y}+9-7\)
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.

\(-3m+9m\)

Try it.

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

Solution

\(-5a\)

Try it.

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

Try it.

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

Solution

\(40x\)

Try it.

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

Try it.

\(-5b-17b\)

Solution

\(-22b\)

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.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Simplify: \(3{x}^{2}+3x+1+8{x}^{2}+5x+5.\)
   If you missed this problem, review .

   Gosi nzaghachi

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

2. Subtract: \((5n+8)-(2n-1).\)
   If you missed this problem, review .

   Gosi nzaghachi

   \(3n+9\)

3. Evaluate: \(4x{y}^{2}\) when \(x=-2\) and \(y=5.\)
   If you missed this problem, review .

   Gosi nzaghachi

   \(-200\)

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

   Gosi nzaghachi

   	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.

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

   Gosi nzaghachi

   ⓐ 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.

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

   Gosi nzaghachi

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

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

   Gosi nzaghachi

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

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

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

   Gosi nzaghachi

   ⓐ \(21{q}^{2}\) ⓑ \(13m{n}^{3}\)

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

   Gosi nzaghachi

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

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

    Gosi nzaghachi

    ⓐ

    	\(\ {a}^{2}+7{b}^{2}-6{a}^{2}\)
    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: ⓐ \(8{y}^{2}+3{z}^{2}-3{y}^{2}\) ⓑ \({m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.\)

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

    	\(({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})\)
    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 \(({a}^{2}+{b}^{2}).\)

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

    	\(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)\)
    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: \((3{x}^{2}-4xy+5{y}^{2})+(2{x}^{2}-xy).\)

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

    ⓐ
    

    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 ⓐ \(f(3)\) ⓑ \(f(-5)\) ⓒ \(f(0).\)

    Gosi nzaghachi

    ⓐ 18 ⓑ 50 ⓒ \(-15\)

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

    Gosi nzaghachi

    ⓐ 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.

    Gosi nzaghachi

    	\(h(t)=-16{t}^{2}+250\)
    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).

    Gosi nzaghachi

    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.

    Gosi nzaghachi

    The height is 31 feet.

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

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

    Gosi nzaghachi

    ⓐ
    

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

    Gosi nzaghachi

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

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

    Gosi nzaghachi

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

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
    ⓔ \(4pq+17\)

    Gosi nzaghachi

    ⓐ 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

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

    Gosi nzaghachi

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

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

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{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.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Add and Subtract Polynomials

1. Determine the degree of polynomials
2. Add and subtract polynomials
3. Evaluate a polynomial function for a given value
4. Add and subtract polynomial functions
5. A
6. A monomial in one variable is a term of the form
7. The
8. The

Questions people ask

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.

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Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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