Multiplying Polynomials
You probably know how to multiply monomials. For example,
2x^{3} Â· 4x^{2} = 8x^{5}.
To multiply a monomial and a polynomial of two or more terms, we apply the
distributive property. For example,
3x(x^{3}  5) = 3x^{4}  15x.
Example 1
Multiplying a monomial and a polynomial
Find the products.
a) 2ab^{2} Â· 3a^{2}b
b) (1)(5  x)
c) (x^{3}  5x + 2)(3x)
Solution
a) 2ab^{2} Â· 3a^{2}b = 6a^{3}b^{3}
b) (1)(5  x) = 5 + x = x  5
c) Each term of x^{3}  5x + 2 is multiplied by 3x:
(x^{3}  5x + 2)(3x) = 3x^{4} + 15x^{2}  6x
Note what happened to the binomial in Example 1(b) when we multiplied it
by 1. If we multiply any difference by 1, we get the same type of result:
1(a  b) = a + b = b  a.
Because multiplying by 1 is the same as taking the opposite, we can write this
equation as
(a  b) = b  a.
This equation says that a  b and b  a are opposites or additive inverses of each
other. Note that the opposite of a b is a  b, not a  b.
To multiply a binomial and a trinomial, we can use the distributive property or
set it up like multiplication of whole numbers.
Example 2
Multiplying a binomial and a trinomial
Find the product (x + 2)(x^{2} + 3x  5).
Solution
We can find this product by applying the distributive property twice. First we
multiply the binomial and each term of the trinomial:
(x + 2)(x^{2} + 3x  5) 
= (x + 2)x^{2} + (x + 2)3x + (x + 2)(5) 
Distributive property. 

= x^{3} + 2x^{2} + 3x^{2} + 6x 
5x  10 
Distributive property. 

= x^{3} + 5x^{2} + x  10 
Combine like terms. 
We could have found this product vertically:

x^{2} 
+ 3x  
5 



x + 
2 


2x^{2} 
+ 6x
 
10 
2(x^{2} + 3x  5) = 2x^{2}
+ 6x  10 
x^{3} + 
2x^{2} 
 5x 

x(x^{2} + 3x  5) = x^{3}
+ 3x^{2}  5x 
x^{3}
+ 
5x^{2} 
+ x  
10 
Add. 
Helpful Hint
Many students find vertical
multiplication easier than applying
the distributive property
twice horizontally. However,
you should learn both
methods because horizontal
multiplication will help you
with factoring by grouping.
