Section 2: Adding and Subtracting Polynomials

Combining Like Terms

In order to combine like terms, the terms must contain exactly the same variables to the same powers. 

Example 1: Simplify the expression by combining like terms.

7x⁵ + 2x² − 3x² + 3 − x⁵

Solution:

Rewrite the expression so that like terms are grouped:

(7x⁵ − x⁵) + (2x² − 3x²) + 3

Add the coefficients for each set of like terms. The variables carry along:

7x⁵ − x⁵ = (7− 1) x⁵ = 6x⁵

2x² − 3x² = (2 − 3)x² = − 1x² = −x²

The simplified form of 7x⁵ + 2x² − 3x² + 3 − x⁵ is 6x⁵ − x² + 3.

Adding and Subtracting Polynomials

Be sure to change the subtraction to addition before combining like terms.

Example 2: Subtract the polynomials.

(4x − 5y² + 2 − x²) −(−2y + 2x² − 3x + 7)

Solution:

Change the division into addition by distributing negative one through the 2nd set of parentheses. The sign of each term in the 2nd parentheses changes.

(4x −5y² + 2 − x²) −(−2y + 2x² − 3x + 7)

= 4x −5y² + 2 − x² + 2y − 2x² + 3x − 7

Rewrite the expression so that like terms are grouped:

(4x + 3x) + ( − x² − 2x²) + (2 − 7) −5y² + 2y

Combine like terms by adding the coefficients.

The simplified form of (4x − 5y² + 2 − x²) −(−2y + 2x² − 3x + 7)

is  7x −3x² − 5 −5y² + 2y .

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