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Add and Subtract Polynomials

Identify polynomials, monomials, binomials, and trinomials

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:

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

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:

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

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

ⓐ \(\ x=3\)
ⓑ \(\ x=-1\)

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.

\(5x+2\)

Solution

binomial

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.

\(10-9x\)

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

Try it.

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

Try it.

\(3x-12\)

Solution

Try it.

\(4y+17\)

Try it.

\(-13\)

Solution

Try it.

\(-22\)

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.

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

Try it.

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

Solution

−8u

Try it.

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

Try it.

\(5a\ +\ 7b\)

Solution

5a + 7b

Try it.

\(8y\ +\ 6z\)

Try it.

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

Solution

−4a −3b

Try it.

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

Try it.

\(18x-2x\)

Solution

16x

Try it.

\(13a-3a\)

Try it.

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

Try it.

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

Try it.

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

Solution

−3x² + 17x − 1

Try it.

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

Try it.

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

Solution

4a² − 7a − 11

Try it.

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

Try it.

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

Solution

4m² − 10m + 2

Try it.

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

Try it.

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

Solution

11z + 8

Try it.

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

Try it.

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

Solution

12s² − 16s + 9

Try it.

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

Try it.

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

Solution

2p³ + p² + 9p + 10

Try it.

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

Try it.

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

Solution

x² + 3x + 4

Try it.

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

Try it.

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.

Try it.

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

ⓐ \(\ y=5\)
ⓑ \(\ y=-2\)
ⓒ \(\ y=0\)

Solution

ⓐ 187
ⓑ 40
ⓒ 2

Try it.

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

ⓐ \(\ y=-4\\)
ⓑ \(\ y=1\)
ⓒ \(y=0\)

Try it.

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

ⓐ \(\ x=3\)
ⓑ \(\ x=0\)
ⓒ \(x=-1\)

Solution

ⓐ −104
ⓑ 4
ⓒ 40

Try it.

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

ⓐ \(\ x=-1\\)
ⓑ \(\ x=0\)
ⓒ \(x=2\)

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.

Practice (40)

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

Simplify: \(8x+3x.\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(11x\)
Subtract: \((5n+8)-(2n-1).\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(3n+9\)
Evaluate: \(4{y}^{2}\) when \(y=5\)
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

\(100\)
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. ⓔ \(\ n\)
Fi àwọn àgbèwọlé hàn

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 whether each polynomial is a monomial, binomial, trinomial, or other polynomial.
1. ⓐ \(\ z\)
2. ⓑ \(\ 2{x}^{3}-4{x}^{2}-x-8\)
3. ⓒ \(\ 6{x}^{2}-4x+1\)
4. ⓓ \(\ 9-4{y}^{2}\)
5. ⓔ \(\ 3{x}^{7}\)
Fi àwọn àgbèwọlé hàn
1. ⓐ monomial
2. ⓑ polynomial
3. ⓒ trinomial
4. ⓓ binomial
5. ⓔ monomial
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.
1. ⓐ \(\ {y}^{3}-8\)
2. ⓑ \(\ 9{x}^{3}-5{x}^{2}-x\)
3. ⓒ \(\ {x}^{4}-3{x}^{2}-4x-7\)
4. ⓓ \(\ \text{-}{y}^{4}\)
5. ⓔ \(\ w\)
Fi àwọn àgbèwọlé hàn
1. ⓐ binomial
2. ⓑ trinomial
3. ⓒ polynomial
4. ⓓ monomial
5. ⓒ monomial

Find the degree of the following polynomials:

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

Fi àwọn àgbèwọlé hàn

ⓐ	\(4x\)
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.

Find the degree of the following polynomials:
1. ⓐ \(\ -6y\)
2. ⓑ \(\ 4x-1\)
3. ⓒ \(\ 3{x}^{4}+4{x}^{2}-8\)
4. ⓓ \(\ 2{y}^{2}+3y+9\)
5. ⓔ \(\ -18\)
Fi àwọn àgbèwọlé hàn
1. ⓐ 1
2. ⓑ 1
3. ⓒ 4
4. ⓓ 2
5. ⓔ 0
Find the degree of the following polynomials:
1. ⓐ \(\ 47\)
2. ⓑ \(\ 2{x}^{2}-8x+2\)
3. ⓒ \(\ {x}^{4}-16\)
4. ⓓ \(\ {y}^{5}-5{y}^{3}+y\)
5. ⓔ \(\ 9{a}^{3}\)
Fi àwọn àgbèwọlé hàn
1. ⓐ 0
2. ⓑ 2
3. ⓒ 4
4. ⓓ 5
5. ⓔ 3
Add: \(17{x}^{2}+6{x}^{2}.\)

Fi àwọn àgbèwọlé hàn

\(17{x}^{2}+6{x}^{2}\)
Combine like terms. \(23{x}^{2}\)
Add: \(12{x}^{2}+5{x}^{2}.\)

Fi àwọn àgbèwọlé hàn

17x²
Add: \(-11{y}^{2}+8{y}^{2}.\)

Fi àwọn àgbèwọlé hàn

−3y²
Subtract: \(11n-(-8n).\)

Fi àwọn àgbèwọlé hàn

\(11n-(-8n)\)
Combine like terms. \(19n\)
Subtract: \(9n-(-5n).\)

Fi àwọn àgbèwọlé hàn

14n
Subtract: \(-7{a}^{3}-(-5{a}^{3}).\)

Fi àwọn àgbèwọlé hàn

−2a³
Simplify: \({a}^{2}+4{b}^{2}-7{a}^{2}.\)

Fi àwọn àgbèwọlé hàn

\({a}^{2}+4{b}^{2}-7{a}^{2}\)
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: \(3{x}^{2}+3{y}^{2}-5{x}^{2}.\)

Fi àwọn àgbèwọlé hàn

−2x² + 3y²
Add: \(2{a}^{2}+{b}^{2}-4{a}^{2}.\)

Fi àwọn àgbèwọlé hàn

−2a² + b²
Find the sum: \((4{x}^{2}-5x+1)+(3{x}^{2}-8x-9).\)

Fi àwọn àgbèwọlé hàn

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.
Find the sum: \((3{x}^{2}-2x+8)+({x}^{2}-6x+2).\)

Fi àwọn àgbèwọlé hàn

4x² − 8x + 10
Find the sum: \((7{y}^{2}+4y-6)+(4{y}^{2}+5y+1).\)

Fi àwọn àgbèwọlé hàn

11y² + 9y − 5
Find the difference: \((7{u}^{2}-5u+3)-(4{u}^{2}-2).\)

Fi àwọn àgbèwọlé hàn

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.
Find the difference: \((6{y}^{2}+3y-1)-(3{y}^{2}-4).\)

Fi àwọn àgbèwọlé hàn

3y² + 3y + 3
Find the difference: \((8{u}^{2}-7u-2)-(5{u}^{2}-6u-4).\)

Fi àwọn àgbèwọlé hàn

3u² − u + 2
Subtract: \(({m}^{2}-3m+8)\) from \((9{m}^{2}-7m+4).\)

Fi àwọn àgbèwọlé hàn

Distribute and identify like terms.
Rearrange the terms.
Combine like terms.
Subtract: \((4{n}^{2}-7n-3)\) from \((8{n}^{2}+5n-3).\)

Fi àwọn àgbèwọlé hàn

4n² + 12n
Subtract: \(({a}^{2}-4a-9)\) from \((6{a}^{2}+4a-1).\)

Fi àwọn àgbèwọlé hàn

5a² + 8a + 8
Evaluate \(3{x}^{2}-9x+7\) when
1. ⓐ \(\ x=3\)
2. ⓑ \(\ x=-1\)
Fi àwọn àgbèwọlé hàn

ⓐ \(x=3\)
\(3{x}^{2}-9x+7\)
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. \(19\)
Evaluate: \(2{x}^{2}+4x-3\) when
1. ⓐ \(\ x=2\)
2. ⓑ \(\ x=-3\)
Fi àwọn àgbèwọlé hàn
1. ⓐ 13
2. ⓑ 3
Evaluate: \(7{y}^{2}-y-2\) when
1. ⓐ \(\ y=-4\)
2. ⓑ \(\ y=0\)
Fi àwọn àgbèwọlé hàn
1. ⓐ 114
2. ⓑ −2
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.

Fi àwọn àgbèwọlé hàn

Substitute 3 for \(t\)
Simplify the expression with the exponent.
Multiply.
Simplify.
The polynomial \(-8{t}^{2}+24t+4\) gives the height, in feet, of a ball \(t\) seconds after it is tossed into the air, from an initial height of \(4\) feet. Find the height after \(t=3\) seconds.

Fi àwọn àgbèwọlé hàn

4 feet
The polynomial \(-8{t}^{2}+24t+4\) gives the height, in feet, of a ball \(x\) seconds after it is tossed into the air, from an initial height of \(4\) feet. Find the height after \(t=2\) seconds.

Fi àwọn àgbèwọlé hàn

20 feet
\({z}^{2}-5z-6\)
\({a}^{2}+9a+18\)

Fi àwọn àgbèwọlé hàn

trinomial
\(-12{p}^{4}\)
\({y}^{3}-8{y}^{2}+2y-16\)

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

monomial
\({m}^{4}+4{m}^{3}+6{m}^{2}+4m+1\)
\(8{a}^{5}-2{a}^{3}+1\)

Fi àwọn àgbèwọlé hàn

5

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.

\approx

approximately equal
Equal to the precision shown, not exactly.

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

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Add and Subtract Polynomials

Identify polynomials, monomials, binomials, and trinomials
Determine the degree of polynomials
Add and subtract monomials
Add and subtract polynomials
Evaluate a polynomial for a given value

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Wárá

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

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