Section 4.4 Multiplying Binomials
INTRODUCTION
In Section 4.3 you were introduced to multiplying two polynomials. You saw how to use the distributive
property to find the product of two binomials and to find the product of a binomial and a trinomial.
This section will focus only on the product of two binomials. However, even though it may seem as though
you’re learning some shortcuts, you’re really only applying the distributive property again and again.
MULTIPLYING BINOMIALS, THE FOIL METHOD
You have learned this fact already, and it’s still very important:
Multiplying two binomials will always result, initially, in four products and four terms.
There is another way to think about getting four terms from multiplying two binomials. It’s really no
different from what you’ve just learned; this new method simply gives us a way to talk about the four products
that result.
When multiplying two binomials, such as (2x + 3)(4x + 5), notice that each binomial has, of course, two
terms: a first term and a second term. We could even say that each binomial has a first term and a last term.
In the product (2x + 3)(4x + 5), the FIRST terms are 2x and 4x and the LAST terms are +3 and +5.
Furthermore, if you look at the order in which the whole product is written, left to right, we have the term 2x
on the far left and the term +5 on the far right. One might look at this and say that 2x and +5 are the OUTER
terms.
Lastly, just in the way it looks, it seems as though the terms +3 and 4x are in the middle of the whole
product; we could say that +3 and 4x are the INNER terms.
A visual representation of this whole idea is presented here.
Multiplying Binomials
page 4.4 - 1
Most math teachers like this method of presentation because it makes it easier for us to talk about the terms
and their products. No one knows who first developed this idea, but whoever it was wrote it in a slightly different
order. This order can make more sense:
F
O
Firsts
I
Outer
(2x + 3)(4x + 5)
L
Inner
(2x + 3)(4x + 5)
(2x + 3)(4x + 5)
Lasts
(2x + 3)(4x + 5)
Notice that, when written in this order, the special pairs of terms spell out the word FOIL. You can use this
acronym to make sure that you’ve got all four of the individual products. Really, it’s most important to know that
Multiplying two binomials will always result, initially, in four products and four terms.
The “FOIL method” doesn’t tell us anything new; it simply gives us a way to speak about the four individual
products that result. The results are the same as when we applied the distributive property.
In multiplying (2x + 3)(4x + 5), we will get four individual products. Based on the FOIL method, those
products are:
2x · 4x = 8x2
8x2
=
2x · 5 = 10x
+
3 · 4x = 12x
10x
+
12x
3 · 5 = 15
+
15
8x2 + 22x + 15
Oftentimes, but not always, the middle two products—the Outer and Inner products—will be like terms and
can be combined.
Example 1:
a)
Find the product using the FOIL method. Be sure to simplify by combining like terms.
(x + 5)(x + 2)
F
O
= x2
+ 2x
I
+ 5x
b)
L
+ 10
= x2 + 7x + 10
(The middle two terms combined.)
Multiplying Binomials
(2x – 3)(6x – 5)
F
O
= 12x2 - 10x
I
- 18x
(x – 5)(2x2 + 4)
c)
L
+ 15
= 12x2 – 28x + 15
(The middle two terms combined.)
F
= 2x3
O
+ 4x
I
L
- 10x2
- 20
= 2x3 – 10x2 + 4x – 20
(The middle two terms could not
combine, and we need to write
the polynomial in descending
order.)
page 4.4 - 2
Exercise 1
Practice using the FOIL method to multiply these binomials. Remember to first write all four
terms, then combine like terms, and write the answer in descending order.
a)
(x + 6)(x + 4)
b)
(3x – 5)(2x + 3)
c)
(3x + 1)(4x + 6)
d)
(x – 9)(x + 9)
e)
(x + 6)(x – 6)
f)
(3x – 7)(3x + 7)
g)
(2x – 1)(3x – 4)
h)
(2x + 3)(2x + 3)
i)
(5x – 2)(5x – 2)
j)
(5x – 2)(6x + 3)
k)
(2x – 1)(4x + 2)
l)
(x2 – 2)(x + 3)
m)
(5x – 3)(x2 – 4)
n)
(8x2 – 1)(x2 + 2)
o)
(3x2 – 4)(3x2 + 4)
Multiplying Binomials
page 4.4 - 3
BINOMIAL PRODUCTS, THE ONE-STEP METHOD
All binomial products have, initially, four terms. As you could see from the previous example (check your
answers if you need to), some of them can be simplified to just two terms, most can be simplified to three terms,
and some cannot be simplified at all.
Of those that can be simplified to three terms—a trinomial—it is almost always because the outer and inner
products are like terms. Knowing this, we can (with practice) reduce the number of steps required to find the
product—the trinomial—by doing some mental mathematics.
Consider the product (2x + 5)(4x – 3), and think about it:
a)
the first product, F, is going to be an x2 term :
8x2 in this case;

This becomes the first term of the trinomial
b)
the outer product, O, and inner product, I,
will be like terms , both will be x terms :
- 6x for the outer product
and, being like terms, they can be combined:
+ 20x for the inner product
______
+ 14x.

This becomes the middle term of the trinomial
c)
the last product, L, is going to be a constant :
- 15 in this case;

This becomes the last term of the trinomial
Putting all of the terms in place, the trinomial is:
8x2 + 14x – 15
The key to doing this—finding the product of two binomials in one step—is being willing and able to do the
calculations mentally. And, thinking is required.
On the next page are many binomial products. These are to be used as practice for the mental exercise of
multiplying binomials in one step. If, at first, you are unable to find the products in just one step, then work them
out—in several steps—on a separate piece of paper and check your answers (at the end of this section). Then try
to do them mentally here.
Finding these products mentally will strengthen your algebraic thinking skills and be of greater benefit for
future sections in this book.
Multiplying Binomials
page 4.4 - 4
Exercise 2
Practice using the FOIL method to multiply these binomials in just one step!. Remember to
write the answer in descending order.
a)
(x + 2)(x + 6)
b)
(x – 5)(x + 3)
c)
(x – 5)(x – 4)
d)
(x – 5)(x + 6)
e)
(x + 7)(x – 2)
f)
(x + 9)(x + 6)
g)
(x – 8)(x – 8)
h)
(x + 5)(x + 5)
i)
(x + 6)(x – 6)
j)
(x – 4)(x + 4)
k)
(3x + 2)(x + 6)
l)
(x – 6)(2x + 1)
m)
(2x + 1)(x + 4)
n)
(3x + 1)(2x + 6)
o)
(4x + 5)(6x + 1)
p)
(2x + 1)(x – 4)
q)
(5x + 2)(5x + 2)
r)
(3x – 5)(x – 2)
s)
(3x – 7)(3x + 7)
t)
(5x – 2)(5x + 2)
u)
(4x + 5)(4x – 5)
Multiplying Binomials
page 4.4 - 5
Answers to each Exercise
Section 4.4
Exercise 1:
Exercise 2:
a)
x2 + 10x + 24
b)
6x2 – x – 15
c)
12x2 + 22x + 6
d)
x2 – 81
e)
x2 – 36
f)
9x2 – 49
g)
6x2 – 11x + 4
h)
4x2 + 12x + 9
i)
25x2 – 20x + 4
j)
30x2 + 3x – 6
k)
8x2 – 2
l)
x3 + 3x2 – 2x – 6
m)
5x3 – 3x2 – 20x + 12
n)
8x4 + 15x2 – 2
o)
9x4 – 16
a)
x2 + 8x + 12
b)
x2 – 2x – 15
c)
x2 – 9x + 20
d)
x2 + x – 30
e)
x2 + 5x – 14
f)
x2 + 15x + 54
g)
x2 – 16x + 64
h)
x2 + 10x + 25
i)
x2 – 36
j)
x2 – 16
k)
3x2 + 20x + 12
l)
2x2 – 11x – 6
m)
2x2 + 9x + 4
n)
6x2 + 20x + 6
o)
24x2 + 34x + 5
p)
2x2 – 7x – 4
q)
25x2 + 20x + 4
r)
3x2 – 11x + 10
s)
9x2 – 49
t)
25x2 – 4
u)
16x2 – 25
Multiplying Binomials
page 4.4 - 6
Section 4.4
1.
Focus Exercises
Practice using the FOIL method to multiply these binomials. Remember to first write all four terms, then
combine like terms, and write the answer in descending order.
a)
(x + 3)(x + 9)
b)
(x – 5)(x – 3)
c)
(x + 1)(x – 8)
d)
(x + 7)(x – 2)
e)
(3x + 5)(2x – 5)
f)
(2x – 9)(4x – 1)
g)
(5x – 2)(5x + 2)
h)
(4x + 3)(4x – 3)
i)
(7x – 1)(7x – 1)
j)
(3x + 6)(3x + 6)
k)
(x – 3)(2x + 8)
l)
(x2 + 4)(x + 5)
m)
(2x – 1)(3x2 – 6)
n)
(4x2 – 5)(x2 + 1)
o)
(2x2 – 5)(2x2 – 5)
Multiplying Binomials
page 4.4 - 7
2.
Practice using the FOIL method to multiply these binomials in just one step!. Remember to write the
answer in descending order.
a)
(x + 8)(x + 2)
b)
(x + 9)(x + 6)
c)
(x – 3)(x – 8)
d)
(x – 6)(x – 4)
e)
(x + 10)(x – 3)
f)
(x + 2)(x – 4)
g)
(x + 6)(x + 6)
h)
(x – 7)(x – 7)
i)
(x + 8)(x – 8)
j)
(x – 10)(x + 10)
k)
(2x + 1)(x + 5)
l)
(4x + 3)(x + 2)
m)
(3x – 6)(2x – 2)
n)
(5x – 4)(2x – 3)
o)
(x + 6)(3x – 5)
p)
(7x + 1)(2x – 3)
q)
(4x – 3)(5x + 1)
r)
(2x – 5)(2x – 5)
s)
(6x + 1)(6x + 1)
t)
(9x – 2)(9x + 2)
u)
(6x + 4)(6x – 4)
Multiplying Binomials
page 4.4 - 8