7.9.2 • Finding Products of Differences
Activity
1. Show that
(
x
−
1
)
(
x
−
1
)
(
x
−
1
)
(
x
−
1
)
x
2
−
2
x
+
1
x
2
−
2
x
+
1
distributive property . Show your reasoning.
Compare your answer:
Drawing a Diagram:
The sum of the partial products are
x
2
+
−
x
+
−
x
+
1
x
2
+
−
x
+
−
x
+
1
x
2
−
2
x
+
1
x
2
−
2
x
+
1
Applying the Distributive Property:
Applying the distributive property to
(
x
−
1
)
(
x
−
1
)
(
x
−
1
)
(
x
−
1
)
x
2
+
x
·
(
−
1
)
+
(
−
1
)
·
x
+
(
−
1
)
2
=
x
2
−
x
−
x
+
1
=
x
2
−
2
x
+
1
x
2
+
x
·
(
−
1
)
+
(
−
1
)
·
x
+
(
−
1
)
2
=
x
2
−
x
−
x
+
1
=
x
2
−
2
x
+
1
2. For each expression, write an equivalent expression. Show your reasoning.
a.
(
x
+
1
)
(
x
−
1
)
(
x
+
1
)
(
x
−
1
)
Compare your answer:
x
2
−
1
x
2
−
1
(
x
+
1
)
(
x
−
1
)
(
x
+
1
)
(
x
−
1
)
x
2
+
x
−
x
−
1
x
2
+
x
−
x
−
1
x
2
−
1
x
2
−
1
b.
(
x
−
2
)
(
x
+
3
)
(
x
−
2
)
(
x
+
3
)
Compare your answer:
x
2
+
x
−
6
x
2
+
x
−
6
(
x
+
3
)
(
x
−
2
)
=
x
2
−
2
x
+
3
x
−
6
=
x
2
+
x
−
6
(
x
+
3
)
(
x
−
2
)
=
x
2
−
2
x
+
3
x
−
6
=
x
2
+
x
−
6
Or using a diagram:
Adding the partial products in the four sub-rectangles gives
x
2
+
3
x
+
−
2
x
+
−
6
x
2
+
3
x
+
−
2
x
+
−
6
x
2
+
x
−
6
x
2
+
x
−
6
c.
(
x
−
2
)
2
(
x
−
2
)
2
Compare your answer:
x
2
−
4
x
+
4
x
2
−
4
x
+
4
(
x
−
2
)
(
x
−
2
)
=
x
2
−
2
x
−
2
x
+
4
=
x
2
−
4
x
+
4
(
x
−
2
)
(
x
−
2
)
=
x
2
−
2
x
−
2
x
+
4
=
x
2
−
4
x
+
4
Video: Finding Products of Differences
Watch the following video to learn more about finding products of
differences.
Which of the following expressions is equivalent to
(
x
−
2
)
2
?
x
2
+
4
x
+
4
x
2
−
4
x
+
4
x
2
+
4
x
2
−
4
Additional Resources
Multiplying Binomials with Negative Numbers
In a previous lesson, you learned how to square a binomial.
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
You can use this to solve the following example.
Example 1
Find the expression equivalent to
(
x
−
3
)
2
(
x
−
3
)
2
This means to multiply
(
x
−
3
)
(
x
−
3
)
(
x
−
3
)
(
x
−
3
)
(
x
−
3
)
2
=
(
x
−
3
)
(
x
−
3
)
(
x
−
3
)
2
=
(
x
−
3
)
(
x
−
3
)
Now, distribute. Remember, you can use FOIL.
Distribute the
x
x
−
3
−
3
Then, combine like terms.
The expression becomes
x
2
−
3
x
−
3
x
+
9
=
x
2
−
6
x
+
9
x
2
−
3
x
−
3
x
+
9
=
x
2
−
6
x
+
9
Example 2
Find the expression equivalent to
(
x
−
4
)
(
x
+
2
)
(
x
−
4
)
(
x
+
2
)
First, distribute. Remember, you can use FOIL.
Multiply the
x
x
(
x
+
2
)
(
x
+
2
)
−
4
−
4
(
x
+
2
)
(
x
+
2
)
x
2
+
2
x
−
4
x
−
8
=
x
2
−
2
x
+
8
x
2
+
2
x
−
4
x
−
8
=
x
2
−
2
x
+
8
Multiplying Binomials with Negative Numbers
Find the expression equivalent to
(
x
+
5
)
(
x
−
3
)
(
x
+
5
)
(
x
−
3
)
Here is how to find an equivalent expression:
Distribute the
x
x
(
x
−
3
)
(
x
−
3
)
5
5
(
x
−
3
)
(
x
−
3
)
x
2
−
3
x
+
5
x
−
15
x
2
−
3
x
+
5
x
−
15
Next, combine like terms.
x
2
+
2
x
−
15
x
2
+
2
x
−
15
