In this section you will learn how to multiply any two polynomials.
Multiplying Monomials with the Product Rule
To multiply two monomials, such as x3 and x5, recall
x3 = x Â· x Â· x and x5 = x Â· x Â· x Â· x Â· x,
The exponent of the product of x3 and x5 is the sum of
the exponents 3 and 5. This example illustrates the product rule for
multiplying exponential expressions.
If a is any real number and m and n are any positive integers, then
am Â· an = am + n.
Find the indicated products.
a) x2 Â· x4 Â· x
c) -4x2y2 Â· 3xy5
|a) x2 Â· x4 Â· x
||= x2 Â· x4 Â· x1
||= (-2)(-3) Â· a Â· a Â· b Â· b
||= (-4)(3)x2 Â· x Â· y2 Â· y5
||= 3a Â· 3a
Be sure to distinguish between adding and multiplying monomials. You can add
like terms to get 3x4 + 2x4 = 5x4, but you
cannot combine the terms in 3w5 + 6w2. However, you can
multiply any two monomials: 3x4 Â· 2x4 = 6x8 and
3w5 Â· 6w2 = 18w7.
To multiply a monomial and a polynomial, we use the distributive property.
Multiply monomials and polynomials
Find each product.
a) 3x2(x3 - 4x)
b) (y2 - 3y + 4)(-2y)
c) -a(b - c)
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.
Use the distributive property to find each product
a) (x + 2)(x + 5)
b) (x + 3)(x2 + 2x - 7)
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.