Multiplying Polynomials

The following are rules regarding the multiplying of variable expressions.

  • Rule 1: To multiply monomials with the same base, keep the base and add the powers:

    x ax b = x a + b

  • Rule 2: To raise a base to a power, keep the base and multiply the powers.

    ( x a ) b = x ab

  • Rule 3: To raise a product to a power, raise each factor in the product to that power.

    ( xy) a = x ay a

Example 1

Simplify each of the following multiplication problems and state which of the preceding rules was applied.

  1. yy 5

  2. ( x 4) 3

  3. (–2 x 4 y 2 z 3) 5

  4. a 3( a 2 b 3) 4

  1. equation

  2. equation

  3. equation

  4. equation

To multiply monomials together, follow this procedure.

  1. Multiply the numerical coefficients together.

  2. Multiply the variables together.

  3. Write the results as a product.

Example 2

Simplify each of the following.

  1. (4 x 2)(3 x 3)

  2. (–8 a 3 b 2)(2 a 2 b 2) 3

  1. (4 x 2)(3 x 3) = (4 × 3)( x 2 x 3) = 12 x 5

  2. (–8 a 3 b 2)(2 a 2 b 2) 3 = (–8 a 3 b 2)(8 a 6 b 6) = –64 a 9 b 8

To multiply polynomials together, multiply each term in one polynomial by each term in the other polynomial. Then simplify if possible.

Example 3

Multiply each of the following.

  1. 5 x(3 x 2 – 4 x + 2)

  2. (4 x – 2)(3 x + 5)

  3. ( x + y)( x 2xy + y 2)

The following shows how each equation is multiplied both horizontally and vertically.

Equation (a) done horizontally: equation

Equation (a) done vertically: equation

Equation (b) done horizontally: equation

Equation (b) done vertically: equation

Equation (c) done horizontally: equation

Equation (c) done vertically: equation