Adding and Subtracting Polynomials

When evaluating a polynomial, we get a real number. So the operations that we perform with real numbers can be performed with polynomials. To add two polynomials, we simply add the like terms.

Example 1

Adding polynomials

Find the sums.

a) (x² - 5x - 7) + (7x² - 4x + 10)

b) (3x³ - 5x² - 7) + (4x² - 2x + 3)

Solution

a) (x² - 5x - 7) + (7x² - 4x + 10) = 8x² - 9x + 3 Combine like terms.

Note that Algebra Solver can easily do all kinds of problems with polynomials that you enter. Click here for a sample screenshot.

b) For illustration we will write this addition vertically:

Helpful Hint

1) When we perform operations with polynomials and write the results as equations, those equations are identities. For example, (2x + 1) + (3x + 7) = 5x + 8 is an identity.

When we subtract polynomials, we subtract like terms. Because a - b = a + (-b), we often perform subtraction by changing signs and adding.

Example 2

Subtracting polynomials

Find the differences.

a) (x² - 7x - 2) - (5x² + 6x - 4)

b) (6y³z - 5yz + 7) - (4y²z - 3yz - 9)

Solution

a) We find the first difference horizontally:

b) For illustration we write (6y³z - 5yz + 7) - (4y²z - 3yz - 9) vertically:

Helpful Hint

For subtraction, write the original problem and then rewrite it as addition with the signs changed.Many students have trouble when they write the original problem and then overwrite the signs. Vertical subtraction is essential for performing long division of polynomials.

It is certainly not necessary to write out all of the steps shown in Examples 1 and 2, but we must use the following rule.

Adding and Subtracting Polynomials

To add two polynomials, add the like terms.

To subtract two polynomials, subtract the like terms.