Multiplying Polynomials
In this section you will learn how to multiply any two polynomials.
Multiplying Monomials with the Product Rule
To multiply two monomials, such as x^{3} and x^{5}, recall
that
x^{3} = x Â· x Â· x and x^{5} = x Â· x Â· x Â· x Â· x,
so
The exponent of the product of x^{3} and x^{5} is the sum of
the exponents 3 and 5. This example illustrates the product rule for
multiplying exponential expressions.
Product Rule
If a is any real number and m and n are any positive integers, then
a^{m} Â· a^{n} = a^{m + n}.
Multiplying monomials
Find the indicated products.
a) x^{2} Â· x^{4} Â· x
b) (2ab)(3ab)
c) 4x^{2}y^{2} Â· 3xy^{5}
d) (3a)^{2}
Solution
a) x^{2} Â· x^{4} Â· x 
= x^{2} Â· x^{4} Â· x^{1} 


= x^{7} 
Product rule 
b) (2ab)(3ab) 
= (2)(3) Â· a Â· a Â· b Â· b 


= 6a^{2}b^{2} 
Product rule 
c) (4x^{2}y^{2})(3xy^{5}) 
= (4)(3)x^{2} Â· x Â· y^{2 }Â· y^{5} 

= 12x^{3}y^{7} 
Product rule 
d) (3a)^{2} 
= 3a Â· 3a 


= 9a^{2} 

Caution
Be sure to distinguish between adding and multiplying monomials. You can add
like terms to get 3x^{4} + 2x^{4} = 5x^{4}, but you
cannot combine the terms in 3w^{5} + 6w^{2}. However, you can
multiply any two monomials: 3x^{4} Â· 2x^{4} = 6x^{8} and
3w^{5} Â· 6w^{2} = 18w^{7}.
Multiplying Polynomials
To multiply a monomial and a polynomial, we use the distributive property.
Multiply monomials and polynomials
Find each product.
a) 3x^{2}(x^{3}  4x)
b) (y^{2}  3y + 4)(2y)
c) a(b  c)
Solution
Note in part c) that either of the last two binomials is the correct answer.
The last one is just a little simpler to read.
Just as we use distributive property to find the product of a monomial and a
polynomial, we can use the distributive property to find the product of two
binomials as the product of a binomial and a trinomial.
Multiplying polynomials
Use the distributive property to find each product
a) (x + 2)(x + 5)
b) (x + 3)(x^{2} + 2x  7)
Solution
a) First multiply each term of x + 5 by x + 2:
b) First multiply each term of the trinomial by x + 3:
Products of polynomials can also be found by arranging the multiplication
vertically like multiplication of whole numbers.
