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# Multiplying a Polynomial by a Monomial

Objective Learn the technique of multiplying a polynomial by a monomial by using the Distributive Property and the properties of exponents.

In this lesson, you will learn to multiply a polynomial by a monomial by using the Distributive Property

The Distributive Property states that a ( b + c ) = ab + ac.

This principle holds even when a , b , and c are monomials, so that b + c represents a polynomial.

Now let's do some examples.

Example 1

Find 2x 3 ( x 2 + 3x + 4).

Solution

Use the Distributive Property to find the product.

 2x 3 ( x 2 + 3x + 4) = 2x 3 (x 2) + 2x 3 (3x) + 2x 3 (4) = 2x 5 + 6x 4 + 8x 3

Now let's see an example in which the polynomial has more than one variable. In this case, you may have to simplify by combining like terms.

Example 2

Find a 2b (3a 3b 2 - 2ab 4 + 3a 2b 2 ).

Solution

Use the Distributive Property to find the product.

 a 2b (3a 3b 2 - 2ab 4 + 3a 2b 2 ) = a 2b (3a 3b 2 ) - a 2b(2ab 4 ) + a 2b (3a 2b 2) = 3a 5b 2 - 2a 3b 5 + 3a 4b 3

## Solving Equations

Sometimes adding, subtracting, and multiplying monomials can be used to help solve a polynomial equation.

Example 3

Solve x( x - 2) + 5 x - 9 = x( x + 1) - 4.

Solution

First multiply each side of the equation by using the Distributive Property.

 x( x - 2) + 5 x - 9 = x( x + 1) - 4 x(x) - x(2) + 5 x - 9 = x(x) + x(1) - 4 x 2 -2x + 5x - 9 = x 2 + x - 4

Now simplify this equation by combining like terms.

 x 2 -2x + 5x - 9 = x 2 + x - 4 x 2 + (-2x + 5x) - 9 = x 2 + x - 4 x 2 + 3x - 9 = x 2 + x - 4 3x - 9 = x - 4 Subtract x 2 from each side. 2x - 9 = - 4 Subtract x from each side. 2x = 5 Add 9 to each side. x = 5/2 Divide each side by 2.

Multiplication of a polynomial by a monomial is an important technique. It will immediately lead to the more general technique of multiplication of polynomials. Therefore, it is very important that you learn this skill well.