Download Section 4.3 Multiplying Polynomials

Section 4.3 Multiplying Polynomials
So far, you have been introduced to the variety of definitions related to polynomials. You have also found the
sum (addition) and difference (subtraction) between two polynomials. Now it is time to learn about the product
(multiplication) of two polynomials.
You have already worked with multiplying some polynomials, but you haven’t yet seen all of the varieties
possible. For example, you have multiplied two monomials together, and you’ve multiplied (distributed) a
monomial through a binomial and a trinomial. Still, each of these, and more, will be explained in this section. As
you shall see, some are easy, some are more challenging. The key to multiplying polynomials is consistency.
THE PRODUCT OF TWO MONOMIALS
The product of two monomials relies on three basic principles:
(i)
The Associative Property of Multiplication
“If the only operation is multiplication, then we can regroup factors (the parts in a
product); in fact, we can even eliminate parentheses altogether.”
(ii)
The Commutative Property of Multiplication
“If the only operation is multiplication, then we can rearrange the factors to be in any
order that works best.”
(iii) The Product Rule of Powers (Exponents)
xa · xb = x a + b
Example 1:
Find the product (multiply these monomials).
(3x4)(2x5)
a)
Procedure
b)
(- 5x2y)(4xy3)
First, remove the parentheses (this is the associative property at work);
Second, rearrange the factors in a way that makes sense (the commutative property);
Third, multiply appropriately.
a)
(3x4)(2x5)
= 3x4·2x5
= 3·2 · x4·x5
=
Better written as
b)
6 · x9
6x9
(- 5x2y)(4xy3)
= - 5x2y·4xy3
= - 5·4 · x2·x · y·y3
=
Better written as
- 20 · x3 · y4
- 20x3y4
Basically, in multiplying monomials, we need to multiply the coefficients together as well as any similar
variable factors. In other words, we could just write:
a)
Multiplying Polynomials
(3x4)(2x5) = 6x9
b)
(- 5x2y)(4xy3) = - 20x3y4
page 4.3 - 1
Exercise 1
Find the product. (You may do these in just one step if you wish.)
a)
(2c3)(8c4)
b)
(- 9x5)(x2)
c)
(5a3b3)(- 2a5b3)
d)
(- 4xy)(- 3x4y)
SQUARING A MONOMIAL: (MONOMIAL)2
Squaring a monomial is fairly simple since it is multiplying one monomial by itself.
Example 2:
Find the product.
(3x4)2
a)
Procedure
(- 5x2y)2
First, remove the parentheses (this is the associative property at work);
Second, rearrange the factors in a way that makes sense (the commutative property);
Third, multiply appropriately.
(3x4)2
= 3x4·3x4
= 3·3 · x4·x4
a)
=
Better written as
Exercise 2
b)
b)
9 · x8
(- 5x2y)2
= - 5x2y·(- 5)x2y
= - 5·(- 5) · x2·x2 · y·y
=
9x8
Better written as
25 · x4 · y2
25x4y2
Find the product.
a)
(5c3)2
b)
(8c4)2
c)
(- 9x5)2
d)
(- 2a5b3)2
e)
(- 4xy)2
f)
(- 3x4y)2
Multiplying Polynomials
page 4.3 - 2
Assuming you did everything correctly in Exercise 2, look back and notice a few things about the answers:
(i)
all the coefficients are positive perfect squares, and
(ii)
all of the exponents are even numbers.
This information will be helpful later on in this section.
THE PRODUCT OF A MONOMIAL AND A POLYNOMIAL
You have seen this as the Distributive Property. In a moment, though, we’re going to see a new way to use
the distributive property. First let’s practice what we already know.
Example 3:
Find the product by distributing the monomial through the polynomial.
(3x2)(4x2 – 2x)
a)
Procedure
a)
b)
(- 5x2)(x2 + 4x – 1)
As we distribute we’ll get a series of monomial products. More steps are shown here than are
necessary.
3x2(4x2 – 2x)
= 3x2·4x2 – 3x2·2x
= 12x4 – 6x3
b)
- 5x2(x2 + 4x – 1)
= -5x2·x2 + -5x2·4x – -5x2·1
= - 5x4 – 20x3 + 5x2
Really, you could multiply these in just one step. You must, though, be careful and be thorough.
Exercise 3
Find the product.
a)
2c(8c4 + 3c2)
b)
- 3x3(x3 – 5x2 + 6x)
c)
2a3(- 4a4 + 3a + 1)
d)
4x2(2x3 – 3x2 + x – 1)
Another way to consider the product of a monomial and a polynomial is to think of something happening
repeatedly. For example, in multiplying 3y by 4y2 + 2y + 5, we need to repeatedly multiply by 3y:
3y(4y2 + 2y + 5)
= 3y·4y2 + 3y·2y + 3y·5
= 12y3 + 6y2 + 15y
In this case, we multiplied 3y by
Multiplying Polynomials
4y2 and 2y and 5. We repeatedly multiplied by 3y.
page 4.3 - 3
Consider, for a moment, a completely different type of action (besides multiplying by 3y) that can be
repeated. For example, consider this unusual circumstance.
In the Johnson family, triplets Josh, Brett and Nate were all born on April 30.
Every year at that time each of the boys gets only two presents, a special gift
from their mom, Laura, and a special gift from their dad, Rob.
To put this scenario in the context of algebra and multiplying polynomials, think of the repeated action of
giving a gift, represented by the k . We might write it like this:
(Laura k ) (Josh & Brett & Nate)
In this case, we wouldn’t be multiplying, but Laura would be distributing gifts (repeatedly).
Laura k Josh & Laura k Brett & Laura k Nate
This is truly the distributive property in action. If we were to use letters instead of names, it might look like this
(notice the use of a plus sign for ‘&’):
L (J + B + N )
= LJ + LB + LN
Continuing with the analogy, let’s look at what happens when we include both of their parents:
(Rob & Laura k) (Josh & Brett & Nate)
What this means is that Rob gives a gift to each of them and Laura gives a gift to each of them. That’s six gifts in
all. In a sense, Rob distributes his gifts and then Laura distributes her gifts.
Rob k Josh & Rob k Brett & Rob k Nate & Laura k Josh & Laura k Brett & Laura k Nate
If we were to look at this more algebraically, we might write:
(R + L) (J + B + N)
= RJ + RB + RN + LJ + LB + LN
The point of all of this is that each member of the first polynomial (the givers) distributes to each member of
the second polynomial (the recipients—the birthday boys). Notice that nowhere do we see that Rob gives a gift to
Laura (it’s not her birthday). We also see that, since there are two givers and three recipients, there are a total of
six gifts.
Let’s put that analogy to work in two actual polynomials, where the repeated action is multiplication.
Consider multiplying (2x + 3) times (6x2 + 4x + 5). In this example, the first polynomial is a binomial, it has
two terms (somewhat like Rob & Laura). The second polynomial is a trinomial, it has three terms (somewhat like
Josh & Brett & Nate).
When we multiply (2x + 3) times (6x2 + 4x + 5) we will get six individual products and six terms (just
like the six gifts). In fact, the first term of the binomial, 2x, will multiply to each term in the trinomial:
2x · 6x2
&
2x · 4x
&
2x · 5
(first three products)
=
Multiplying Polynomials
12x3
&
8x2
&
10x
(first three new terms)
page 4.3 - 4
Likewise, the second term of the binomial, 3, will also multiply to each term in the trinomial:
3 · 6x2
&
3 · 4x
&
3·5
18x2
&
12x
&
15
=
(second three products)
(second three new terms)
Putting all of the terms together, we get six terms in all:
first three new terms
12x3
second three new terms
& 8x2 & 10x
18x2 & 12x & 15
&
Notice that, of the six new terms, some of them are like terms and eventually can be combined. Also, just as Rob
doesn’t give Laura a gift, we never multiply the individual terms of the binomial, the 2x and the 3, together.
Let’s put this into algebraic action.
Example 4:
Multiply these polynomials. Combine like terms, and write the answer in descending order.
(2x + 3)(6x2 + 4x + 5)
Procedure:
Multiply each term in the first polynomial by each term in the second polynomial. It’s
helpful to decide how many terms the product will have after multiplying (and before
combining like terms).
Since there is a binomial (2 terms) and a trinomial (3 terms), then initially there will be six
individual products which become six individual terms.
(2x + 3)(6x2 + 4x + 5)
Six products:
= 2x·6x2 + 2x·4x + 2x·5
+
The six new terms:
= 12x3 +
+
Gathering like terms:
= 12x3
Combining like terms:
= 12x3 + 26x2 + 22x + 15
8x2
+
+ 10x
8x2 + 18x2
3·6x2 + 3·4x + 3·5
18x2 + 12x
+
10x + 12x
+ 15
+ 15
Here is a diagram of the distribution process. Notice that each term of the binomial is multiplied to each terms
of the trinomial.
(2x + 3)(6x 2+ 4x + 5)
Multiplying Polynomials
page 4.3 - 5
Here is a diagram of the distribution process for the product of two binomials. Initially, there will be four
products.
This diagram is shown so that you can begin to do the first step mentally. Knowing that “the sign in front of a
term belongs to the term,” the four products are
2x·6x2
&
2x·(-5x)
&
(-3)·(6x2)
&
(-3)·(-5x)
=
12x3
&
- 10x2
&
- 18x2
&
+ 15x
=
12x3
first step:
Example 5:
a)
–
10x2
– 18x2
+
 This should be the real first step.
15x
Multiply. Combine like terms, and write the answer in descending order.
(3x + 5)(2x + 1)
b)
(2x – 3)(6x2 – 5x)
(x – 5)(2x2 – 4)
c)
Procedure:
Because each product consists of two binomials (2 terms each), the initial product should
have four terms. Think through the diagram above and multiply directly.
Answer:
a)
(3x + 5)(2x + 1)
The four products:
= 6x2 +
Combining like terms:
=
3x
+ 10x
+
5
6x2 + 13x + 5
(2x – 3)(6x2 – 5x)
b)
Be sure to recognize the sign in front of each term.
The four products:
=
Combining like terms:
=
12x3
–
10x2
–
18x2
+
15x
12x3 – 28x2 + 15x
(x – 5)(2x2 – 4)
c)
This product is a little different because, as it turns out, none of the terms are like terms.
Four products:
=
2x3
In descending order:
=
2x3 – 10x2 – 4x + 20
Multiplying Polynomials
–
4x
–
10x2
+
20
page 4.3 - 6
Exercise 4
Find each product. Combine like terms, and write the answer in descending order.
a)
(x + 3)(x + 8)
b)
(x – 2)(x – 6)
c)
(3x + 4)(x – 2)
d)
(3x – 1)(4x – 3)
e)
(x2 + 5)(x – 3)
f)
(2x2 – 3)(5x2 – 1)
g)
(x + 2)(x2 + 3x + 1)
h)
(x – 2)(2x2 – x + 5)
i)
(4x + 2)(x2 – 5x – 1)
j)
(3x – 2)(5x2 – 4x + 6)
Multiplying Polynomials
page 4.3 - 7
This is important to know:
Multiplying two binomials will always result, initially, in four products and four terms.
Sometimes the four terms can simplify to just three terms because two of them are like terms; in fact, it’s
possible that the product of two binomials eventually simplifies to just two terms.
Example 6:
Multiply and simplify.
a)
Procedure:
(x + 5)(x – 5)
b)
(2x – 3)(2x + 3)
Multiply as before. Be sure you first get four products and four terms. You may do that part
of the process mentally.
a)
(x + 5)(x – 5)
b)
(2x – 3)(2x + 3)
= x·x – x·5 + 5·x – 5·5
= 2x·2x + 2x·3 – 3·2x – 3·3
= x2 – 5x + 5x – 25
= 4x2 + 6x – 6x – 9
= x2 +
= 4x2 +
0x
– 25
= x2 – 25
0x
– 9
= 4x2 – 9
In both cases, the two like terms are exact opposites and add to just 0.
Exercise 5
Find each product. Combine like terms, and write the answer in descending order.
a)
(x + 2)(x – 2)
b)
(y – 4)(y + 4)
c)
(6d + 7)(6d – 7)
d)
(3w – 5)(3w + 5)
e)
(9y + 1)(9y – 1)
f)
(3x – 8)(3x + 8)
Multiplying Polynomials
page 4.3 - 8
Answers to each Exercise
Section 4.3
Exercise 1:
a)
16c7
b)
- 9x7
c)
- 10a8b6
d)
12x5y2
Exercise 2:
a)
25c6
b)
64c8
c)
81x10
d)
4a10b6
e)
16x2y2
f)
9x8y2
a)
16c5 + 6c3
b)
- 3x6 + 15x5 – 18x4
c)
- 8a7 + 6a4 + 2a3
d)
8x5 – 12x4 + 4x3 – 4x2
a)
x2 + 11x + 24
b)
x2 – 8x + 12
c)
3x2 – 2x – 8
d)
12x2 – 13x + 3
e)
x3 – 3x2 + 5x – 15
f)
10x4 – 17x2 + 3
g)
x3 + 5x2 + 7x + 2
h)
2x3 – 5x2 + 7x – 10
i)
4x3 – 18x2 – 14x – 2
j)
15x3 – 22x2 + 26x – 12
a)
x2 – 4
b)
y2 – 16
c)
36d2 – 49
d)
9w2 – 25
e)
81y2 – 1
f)
9x2 – 64
Exercise 3:
Exercise 4:
Exercise 5:
Multiplying Polynomials
page 4.3 - 9
Section 4.3
1.
Find the product. (You may do these in just one step if you wish.)
a)
(3y2)(- 2y3)
b)
(- x4)(2x)
c)
(3ab)(- a6b2)
d)
(5y4)2
e)
(- 3c2)2
f)
(- p 3 )2
g)
(6a4b)2
h)
(wxy)2
i)
(- 4x3y5)2
2.
3.
Focus Exercises
Find the product. Be sure to write the result in descending order.
a)
3y(2y5 – y2)
b)
- 8x2(x – 5 + 2x2)
c)
- a3 (- a3 – 5 + a)
d)
2x2(3x – 9 + x3 – 6x2)
Find each product. Combine like terms, and write the answer in descending order.
a)
(x + 4)(x + 7)
b)
(x – 8)(x – 5)
c)
(x + 3)(x – 12)
d)
(x – 2)(x + 6)
Multiplying Polynomials
page 4.3 - 10
4.
Find each product. Combine like terms, and write the answer in descending order.
a)
(4x + 2)(x – 3)
b)
(9x – 2)(3x + 1)
c)
(- 2x – 3)(5x – 4)
d)
(x2 – 1)(x2 – 6)
e)
(4x + 3)(x2 – 3x – 2)
f)
(x2 – 3)(2x + 3 – x2)
g)
(x – 6)(x + 6)
h)
(y + 5)(y – 5)
i)
(2p + 9)(2p – 9)
j)
(- 4m + 1)(- 4m – 1)
k)
(4x2 – 3)(4x2 + 3)
l)
(5x3 – 1)(5x3 + 1)
Multiplying Polynomials
page 4.3 - 11