4.4 Adding, Subtracting, and Evaluating Polynomials

Learning Objectives

Add and subtract polynomials by combining like terms.
Evaluate a polynomial for a given value of its variable.

Adding and subtracting polynomials works just like combining like terms — we add or subtract the coefficients of matching terms.

Adding Polynomials

When adding, simply combine like terms.

Example 4.4.1

Simplify: $(3x^2+5x+7)+(2x^2+4x+1)$

Solution

Combine like terms:

$(3x^2+2x^2)+(5x+4x)+(7+1)=5x^2+9x+8$

Example 4.4.2

Simplify: $(4y^3-2y+6)+(3y^3+y-5)$

Solution

$(4y^3+3y^3)+(-2y+y)+(6-5)=7y^3-y+1$

Try It

Simplify each.

Subtracting Polynomials

When subtracting, distribute the negative sign to every term in the second polynomial before combining like terms.

Example 4.4.3

Simplify: $(5x^2+3x-4)-(2x^2+x-6)$

Solution

$5x^2+3x-4-2x^2-x+6$

$(5x^2-2x^2)+(3x-x)+(-4+6)=3x^2+2x+2$

Example 4.4.4

Simplify: $(6m^3-m^2+5)-(-2m^3+3m^2-4)$

Solution

Distribute the negative to the second polynomial:

$6m^3-m^2+5+2m^3-3m^2+4$

$(6m^3+2m^3)+(-m^2-3m^2)+(5+4)=8m^3-4m^2+9$

Try It

Simplify.

Adding and Subtracting Multiple Polynomials

The same rules apply when more than two polynomials are involved.

Example 4.4.5

Simplify: $(x^2+2x-3)+(4x^2-x+1)+(-2x^2+3x-5)$

Solution

Clear parentheses and group like terms:

$x^2+2x-3+4x^2-x+1-2x^2+3x-5$

$(x^2+4x^2-2x^2)+(2x-x+3x)+(-3+1-5)$

$3x^2+4x-7$

Final Answer: $3x^2+4x-7$

Tip: It can help to rewrite the expression so like terms are next to each other (and be especially careful with signs).

Example 4.4.6

Simplify: $(3x^2-4x+5)-(2x^2-3x+7)$

Solution

Distribute the negative:

$3x^2-4x+5-2x^2+3x-7$

$(3x^2-2x^2)+(-4x+3x)+(5-7)=x^2-x-2$

Try It

Simplify.

Evaluating Polynomials

To evaluate a polynomial, substitute the given number for the variable and simplify.

Example 4.4.7

Evaluate $2x^2+3x-4$ when $x=2$ .

Solution

$2(2^2)+3(2)-4=8+6-4=10$

Example 4.4.8

Evaluate $x^3-2x^2+5$ when $x=-1$ .

Solution

$(-1)^3-2(-1)^2+5=-1-2+5=2$

Example 4.4.9

Evaluate $-3x^2+4x-7$ when $x=-3$ .

Solution

$-3(-3)^2+4(-3)-7=-3(9)-12-7=-46$

Example 4.4.10

Evaluate $x^3-2x^2+x-1$ when $x=-2$ .

Solution

$(-2)^3-2(-2)^2+(-2)-1=-8-8-2-1=-19$

Try It

Evaluate each expression.

Key Takeaways

When adding or subtracting, combine coefficients of like terms.
Evaluate by substituting and simplifying step by step.

License

Icon for the Creative Commons Attribution-NonCommercial 4.0 International License

Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

Share This Book