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 x3 and x5, recall
that
x3 = x · x · x and x5 = x · x · x · x · x,
so
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.
Product Rule
If a is any real number and m and n are any positive integers, then
am · an = am + n.
Multiplying monomials
Find the indicated products.
a) x2 · x4 · x
b) (-2ab)(-3ab)
c) -4x2y2 · 3xy5
d) (3a)2
Solution
| a) x2 · x4 · x |
= x2 · x4 · x1 |
|
| |
= x7 |
Product rule |
| b) (-2ab)(-3ab) |
= (-2)(-3) · a · a · b · b |
|
| |
= 6a2b2 |
Product rule |
| c) (-4x2y2)(3xy5) |
= (-4)(3)x2 · x · y2 · y5 |
| |
= -12x3y7 |
Product rule |
| d) (3a)2 |
= 3a · 3a |
|
| |
= 9a2 |
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Caution
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 · 2x4 = 6x8 and
3w5 · 6w2 = 18w7.
Multiplying Polynomials
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)
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)(x2 + 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.
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