© 2016 Shmoop University, Inc. All rights reserved.


At a Glance - Multiplication of a Monomial and a Polynomial

The easiest case of polynomial multiplication is multiplying a monomial and a polynomial. In this case, we "distribute" the monomial to each term in the polynomial. We really do distribute it, though. We weren't using the quotation marks to be sarcastic or ironic.

Sample Problem

What is (2x)(3x2 + 4x + 9)?

We use the distributive property to distribute (2x) over the longer polynomial, then simplify the resulting terms.

(2x)(3x2 + 4x + 9) = (2x)(3x2) + (2x)(4x) + (2x)(9)
  = 6x3 + 8x2 + 18x

Sample Problem

What is (5x2)(4x3 + 7x)?

We distribute (5x2) over both terms in the second set of parentheses, then simplify.

(5x2)(4x3) + (5x2)(7x) =
20x5 + 35x3

Example 1

What is (2y)(x2 + 3xyy)?

Exercise 1

Find the product of (4x)(x + 11).

Exercise 2

Find the product of (2x3)(x3 + 3x2 + 4x + 9).

Exercise 3

Find the product of (-x2)(2x4 – 11x + 17).

Exercise 4

Find the product of (4x)(x2 + xy + 3y2).

Exercise 5

Find the product of (0)(14x23 + 6x17 – 8x9 + 14).

People who Shmooped this also Shmooped...