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5.2: Add and Subtract Polynomials

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Learning Objectives

By the end of this section, you will be able to:

  • Determine the degree of polynomials
  • Add and subtract polynomials
  • Evaluate a polynomial function for a given value
  • Add and subtract polynomial functions
Note

Before you get started, take this readiness quiz.

  1. Simplify: 3x2+3x+1+8x2+5x+5.
    If you missed this problem, review [link].
  2. Subtract: (5n+8)(2n1).
    If you missed this problem, review [link].
  3. Evaluate: 4xy2 when x=2x and y=5..
    If you missed this problem, review [link].

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 axm, where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and 4a2b3c2.

Definition: MONOMIAL

A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form axm, where a is a constant and m is a whole number.

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.

Definition: 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.

Here are some examples of polynomials.

Polynomial y+1 4a27ab+2b2 4x4+x3+8x29x+1  
Monomial 14 8y2 9x3y5 13a3b2c
Binomial a+7ba+7b 4x2y2 y216 3p3q9p2q
Trinomial x27x+12 9m2+2mn8n2 6k4k3+8k z4+3z21

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.

Definition: 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.

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. Let's start by looking at a monomial. The monomial 8ab2 has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, 1+2,1+2, is 3 so the degree is 3.

CNX_IntAlg_Figure_05_01_001_img_new.jpg

Here are some additional examples.

CNX_IntAlg_Figure_05_01_002_img_new.jpg

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 of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 5.2.1

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

  1. 7y25y+3
  2. 2a4b2
  3. 3x54x36x2+x8
  4. 2y8xy3
  5. 15
Answer
Polynomial Number of terms Type Degree of terms Degree of polynomial
7y25y+3 3 Trinomial 2, 1, 0 2
2a4b22a4b2 1 Monomial 4, 2 6
3x54x36x2+x8 5 Polynomial 5, 3, 2, 1, 0 5
2y8xy3 2 Binomial 1, 4 4
15 1 Monomial 0 0
Example 5.2.2

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

  1. 5
  2. 8y37y2y3
  3. 3x2y5xy+9xy3
  4. 81m24n2
  5. 3x6y3z
Answer a

monomial, 0

Answer b

polynomial, 3

Answer c

trinomial, 3

Answer d

binomial, 2

Answer b

monomial, 10

Example 5.2.3

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

  1. 64k38
  2. 9m3+4m22
  3. 56
  4. 8a47a3b6a2b24ab3+7b4
  5. p4q3
Answer

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

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 5.2.4

Add or subtract:

  1. 25y2+15y2
  2. 16pq3(7pq3).
Answer a

25y2+15y2Combine like terms.40y2

Answer b

16pq3(7pq3)Combine like terms.23pq3

Example 5.2.5

Add or subtract:

  1. 12q2+9q2
  2. 8mn3(5mn3).
Answer

21q213mn3

Example 5.2.6

Add or subtract:

  1. 15c2+8c2
  2. 15y2z3(5y2z3)
Answer

7c210y2z3

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

Example 5.2.7

Simplify:

  1. a2+7b26a2
  2. u2v+5u23v2
Answer

ⓐ Combine like terms.

a2+7b26a2=5a2+7b2

ⓑ There are no like terms to combine. In this case, the polynomial is unchanged.

u2v+5u23v2

Example 5.2.8

Add:

  1. 8y2+3z23y2
  2. m2n28m2+4n2
Answer

5y2+3z2
m2n28m2+4n2

Example 5.2.9

Add:

  1. 3m2+n27m2
  2. pq26p5q2
Answer

4m2+n2
pq26p5q2

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 5.2.10

Find the sum: (7y22y+9)+(4y28y7).

Answer

Identify like terms.(7y2__2y_+9)+(4y2__8y_7)Rewrite without the parentheses,rearranging to get the like terms together.7y2+4y2__2y8y_+97Combine like terms.11y210y+2

Example 5.2.11

Find the sum: (7x24x+5)+(x27x+3)

Answer

8x211x+8

Example 5.2.12

Find the sum: (14y2+6y4)+(3y2+8y+5)

Answer

17y2+14y+1

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

Example 5.2.13

Find the difference: (9w27w+5)(2w24)

Answer

(9w27w+5)(2w24)Distribute and identify like terms.9w2__7w_+52w2__+4Rearrange the terms.9w22w2__7w_+5+4Combine like terms.7w27w+9

Example 5.2.14

Find the difference: (8x2+3x19)(7x214)

Answer

x2+3x5

Example 5.2.15

Find the difference: (9b25b4)(3b25b7)

Answer

6b2+3

Example 5.2.16

Subtract (p2+10pq2q2) from (p2+q2).

Answer

(p2+q2)(p2+10pq2q2)Distribute and identify like terms.p2__+q2_p2__10pq+2q2_Rearrange the terms, putting like terms together.p2p2__10pq+q2+2q2_Combine like terms.10pq+3q2

Example 5.2.17

Subtract (a2+5ab6b2) from (a2+b2)

Answer

5ab+7b2

Example 5.2.18

Subtract (m27mn3n2) from (m2+n2).

Answer

7mn+4n^2

Example 5.2.19

Find the sum: (u26uv+5v2)+(3u2+2uv)

Answer

(u26uv+5v2)+(3u2+2uv)Distribute and identify like terms.u2__6uv_+5v2+3u2__+2uv_Rearrange the terms to put like terms together.u2__+3u2__6uv_+2uv_+5v2Combine like terms.4u24uv+5v2

Example 5.2.20

Find the sum: (3x24xy+5y2)+(2x2xy)

Answer

5x25xy+5y2

Example 5.2.21

Find the sum: (2x23xy2y2)+(5x23xy)

Answer

7x26xy2y2

When we add and subtract more than two polynomials, the process is the same.

Example 5.2.22

Simplify: (a3a2b)(ab2+b3)+(a2b+ab2)

Answer

(a3a2b)(ab2+b3)+(a2b+ab2)Distributea3a2bab2b3+a2b+ab2Rearrange the terms to put like terms together.a3a2b+a2bab2+ab2b3Combine like terms.a3b3

Example 5.2.23

Simplify: (x3x2y)(xy2+y3)+(x2y+xy2)

Answer

x3+y3

Example 5.2.24

Simplify: (p3p2q)+(pq2+q3)(p2q+pq2)

Answer

p33p2q+q3

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, f(x)=x2+5x+6 and g(x)=3x4 are polynomial functions, because x2+5x+6 and 3x4 are polynomials.

Definition: POLYNOMIAL FUNCTION

A polynomial function is a function whose range values are defined by a polynomial.

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 5.2.25

For the function f(x)=5x28x+4 find:

  1. f(4)
  2. f(2)
  3. f(0).
Answer

  .
.   .
Simplify the exponents. .
Multiply. .
Simplify. .

  .
. .
Simplify the exponents. .
Multiply. .
Simplify. .

  .
.   .
Simplify the exponents. .
Multiply. .
Example 5.2.26

For the function f(x)=3x2+2x15, find

  1. f(3)
  2. f(5)
  3. f(0).
Answer

ⓐ 18 ⓑ 50 ⓒ 15

Example 5.2.27

For the function g(x)=5x2x4, find

  1. g(2)
  2. g(1)
  3. g(0).
Answer

ⓐ 20 ⓑ 2 ⓒ 4

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 5.2.28

The polynomial function h(t)=16t2+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.

Answer

h(t)=16t2+250To find h(2), substitute t=2.h(2)=16(2)2+250Simplify.h(2)=16·4+250Simplify.h(2)=64+250Simplify.h(2)=186After 2 seconds the height of the ball is 186 feet.

Example 5.2.29

The polynomial function h(t)=16t2+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).

Answer

The height is 150 feet.

Example 5.2.30

The polynomial function h(t)=16t2+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.

Answer

The height is 31 feet.

Add and Subtract Polynomial Functions

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

Definition: ADDITION AND SUBTRACTION OF POLYNOMIAL FUNCTIONS

For functions f(x) and g(x),

(f+g)(x)=f(x)+g(x)

(fg)(x)=f(x)g(x)

Example 5.2.31

For functions f(x)=3x25x+7 and g(x)=x24x3, find:

  1. (f+g)(x)
  2. (f+g)(3)
  3. (fg)(x)
  4. (fg)(2).
Answer

  .
. .
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)=4x29x+4To find (f+g) (3), substitute x=3.(f+g)(3)=4(3)29·3+4(f+g)(3)=4·99·3+4(f+g)(3)=3627+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. .

.
Example 5.2.32

For functions f(x)=2x24x+3 and g(x)=x22x6, find: ⓐ (f+g)(x)(f+g)(3)(fg)(x)(fg)(2).

Answer

(f+g)(x)=3x26x3

(f+g)(3)=6

(fg)(x)=x22x+9

(fg)(2)=17

Example 5.2.33

For functions f(x)=5x24x1 and g(x)=x2+3x+8, find ⓐ (f+g)(x)(f+g)(3)(fg)(x)(fg)(2).

Answer

(f+g)(x)=6x2x+7

(f+g)(3)=58

(fg)(x)=4x27x9

(fg)(2)=21

Access this online resource for additional instruction and practice with adding and subtracting polynomials.

  • Adding and Subtracting Polynomials

Key Concepts

  • Monomial
    • A monomial is an algebraic expression with one term.
    • A monomial in one variable is a term of the form axm,axm, 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.

Glossary

binomial
A binomial is a polynomial with exactly two terms.
degree of a constant
The degree of any constant is 0.
degree of a polynomial
The degree of a polynomial is the highest degree of all its terms.
degree of a term
The degree of a term is the sum of the exponents of its variables.
monomial
A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form axm,axm, where a is a constant and m is a whole number.
polynomial
A monomial or two or more monomials combined by addition or subtraction is a polynomial.
standard form of a polynomial
A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.
trinomial
A trinomial is a polynomial with exactly three terms.
polynomial function
A polynomial function is a function whose range values are defined by a polynomial.

This page titled 5.2: Add and Subtract Polynomials is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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