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.
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