How to Add and Subtract Polynomials

Learn how to add and subtract polynomials and how to combine like terms and simplify polynomials in this article.

Step by step guide to solve adding and subtracting polynomials

• Adding polynomials is just a matter of combining like terms, with some order of operations considerations thrown in.
• Be careful with the minus signs, and don’t confuse addition and multiplication!

Adding and Subtracting Polynomials – Example 1:

Simplify the expressions. $$(2x^3-4x^4 )-(2x^4-6x^3 )=$$

Solution:

First use Distributive Property for $$-(2x^4-6x^3 )= -2x^4+6x^3$$
$$(2x^3-4x^4 )-(2x^4-6x^3 )=2x^3-4x^4-2x^4+6x^3$$
Now combine like terms: $$2x^3-4x^4-2x^4+6x^3=-6x^4+8x^3$$

Adding and Subtracting Polynomials – Example 2:

Add expressions. $$(x^3-2)+(5x^3-3x^2 )=$$

Solution:

Remove parentheses: $$(x^3-2)+(5x^3-3x^2 )=x^3-2+5x^3-3x^2$$
Now combine like terms: $$x^3-2+5x^3-3x^2=6x^3-3x^2-2$$

Adding and Subtracting Polynomials – Example 3:

Subtract. $$(4x^3+3x^4 )-(x^4-5x^3 )=$$

Solution:

First use Distributive Property for $$-(x^4-5x^3 ), → -(x^4-5x^3 )=-x^4+5x^3$$
$$(4x^3+3x^4 )-(x^4-5x^3 )=4x^3+3x^4-x^4+5x^3$$
Now combine like terms: $$4x^3+3x^4-x^4+5x^3=2x^4+9x^3$$

Adding and Subtracting Polynomials – Example 4:

Add expressions. $$(2x^3-6)+(9x^3-4x^2 )=$$

Solution:

Remove parentheses: $$(2x^3-6)+(9x^3-4x^2 )=2x^3-6+9x^3-4x^2$$
Now combine like terms: $$2x^3-6+9x^3-4x^2=11x^3-4x^2-6$$

Exercises for Adding and Subtracting Polynomials

Simplify each expression.

1. $$\color{blue}{(2x^3 – 2) + (2x^3 + 2)}$$
2. $$\color{blue}{(4x^3 + 5) – (7 – 2x^3)}$$
3. $$\color{blue}{(4x^2 + 2x^3) – (2x^3 + 5)}$$
4. $$\color{blue}{(4x^2 – x) + (3x – 5x^2)}$$
5. $$\color{blue}{(7x + 9) – (3x + 9)}$$
6. $$\color{blue}{(4x^4 – 2x) – (6x – 2x^4)}$$

1. $$\color{blue}{4x^3 }$$
2. $$\color{blue}{6x^3 – 2}$$
3. $$\color{blue}{4x^2 – 5}$$
4. $$\color{blue}{– x^2 + 2x}$$
5. $$\color{blue}{4x}$$
6. $$\color{blue}{6x^4 – 8x}$$

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