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^{ 2}b (3a ^{3}b^{ 2}  2ab^{
4} + 3a^{ 2}b^{ 2} ).
Solution
Use the Distributive Property to find the product.
a^{ 2}b (3a ^{3}b^{ 2}
 2ab^{ 4} + 3a^{ 2}b^{ 2} ) 
= a^{ 2}b (3a ^{3}b^{
2 })  a^{ 2}b(2ab^{ 4} ) + a^{ 2}b
(3a^{ 2}b^{ 2}) 

= 3a^{ 5}b^{ 2}  2a^{
3}b^{ 5} + 3a^{ 4}b ^{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.
