# 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 (2*x*)(3*x*^{2} + 4*x* + 9)?

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

(2x)(3x^{2} + 4x + 9) | = | (2x)(3x^{2}) + (2x)(4x) + (2x)(9) |

= | 6x^{3} + 8x^{2} + 18x |

### Sample Problem

What is (5*x*^{2})(4*x*^{3} + 7*x*)?

We distribute (5*x*^{2}) over both terms in the second set of parentheses, then simplify.

(5*x*^{2})(4*x*^{3}) + (5*x*^{2})(7*x*) =

20*x*^{5} + 35*x*^{3}

#### Exercise 1

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

#### Exercise 2

Find the product of (2*x*^{3})(*x*^{3} + 3*x*^{2} + 4*x* + 9).

#### Exercise 3

Find the product of (-*x*^{2})(2*x*^{4} – 11*x* + 17).

#### Exercise 4

Find the product of (4*x*)(*x*^{2} + *xy* + 3*y*^{2}).

#### Exercise 5

Find the product of (0)(14*x*^{23} + 6*x*^{17} – 8*x*^{9} + 14).