Finding Products of Differences

7.9.2 Finding Products of Differences

7.9.2 • Finding Products of Differences

Activity

1. Show that $(x - 1) (x - 1)$ and $x^{2} - 2 x + 1$ are equivalent expressions by drawing a diagram or applying the distributive property. Show your reasoning.

Compare your answer:

Drawing a Diagram:

	$x$	$- 1$
$x$	$x^{2}$	$- x$
$- 1$	$- x$	$1$

The sum of the partial products are $x^{2} + - x + - x + 1$ or $x^{2} - 2 x + 1$ .

Applying the Distributive Property:

Applying the distributive property to $(x - 1) (x - 1)$ gives $x^{2} + x \cdot (- 1) + (- 1) \cdot x + (- 1)^{2} = x^{2} - x - x + 1 = x^{2} - 2 x + 1$ .

2. For each expression, write an equivalent expression. Show your reasoning.

a. $(x + 1) (x - 1)$

Compare your answer:

$x^{2} - 1$ . For example, applying the distributive property to $(x + 1) (x - 1)$ gives $x^{2} + x - x - 1$ , which equals $x^{2} - 1$ .

b. $(x - 2) (x + 3)$

Compare your answer:

$x^{2} + x - 6$ , because $(x + 3) (x - 2) = x^{2} - 2 x + 3 x - 6 = x^{2} + x - 6$ .

Or using a diagram:

	$x$	$3$
$x$	$x^{2}$	$3 x$
$- 2$	$- 2 x$	$- 6$

Adding the partial products in the four sub-rectangles gives $x^{2} + 3 x + - 2 x + - 6$ , which equals $x^{2} + x - 6$ .

c. $(x - 2)^{2}$

Compare your answer:

$x^{2} - 4 x + 4$ , because $(x - 2) (x - 2) = x^{2} - 2 x - 2 x + 4 = x^{2} - 4 x + 4$ .

Video: Finding Products of Differences

Watch the following video to learn more about finding products of differences.

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Self Check

Which of the following expressions is equivalent to

(x-2)^2

$x^2+4x+4$
$x^2-4x+4$
$x^2+4$
$x^2-4$

Additional Resources

Multiplying Binomials with Negative Numbers

In a previous lesson, you learned how to square a binomial.

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$ and $(a - b)^{2} = a^{2} - 2 a b + b^{2}$

You can use this to solve the following example.

Example 1

Find the expression equivalent to $(x - 3)^{2}$ .

This means to multiply $(x - 3)$ by $(x - 3)$ . So, $(x - 3)^{2} = (x - 3) (x - 3)$ .

Now, distribute. Remember, you can use FOIL.

A diagram showing the quantity of x minus 3 times the quantity of x minus 3. On the top, a blue arrow is going from x in the first binomial to x in the second binomial. Another blue arrow is going from x in the first binomial to negative 3 in the second binomial. On the bottom, in red, an arrow is going from negative 3 in the first binomial to x in the second binomial and then another arrow is going from negative 3 in the first binomial to negative 3 in the second binomial.

Distribute the $x$ to the second binomial, then distribute the $- 3$ to the second binomial.

Then, combine like terms.

The expression becomes $x^{2} - 3 x - 3 x + 9 = x^{2} - 6 x + 9$ .

Example 2

Find the expression equivalent to $(x - 4) (x + 2)$ .

First, distribute. Remember, you can use FOIL.

A diagram showing the quantity of x minus 4 times the quantity of x plus 2. On the top, a blue arrow is going from x in the first binomial to x in the second binomial. Another blue arrow is going from x in the first binomial to 2 in the second binomial. On the bottom, in red, an arrow is going from negative 4 in the first binomial to x in the second binomial and then another arrow is going from negative 4 in the first binomial to 2 in the second binomial.

Multiply the $x$ by $(x + 2)$ , then multiply the $- 4$ by $(x + 2)$ and combine like terms:

$x^{2} + 2 x - 4 x - 8 = x^{2} - 2 x + 8$

Try it

Multiplying Binomials with Negative Numbers

Find the expression equivalent to $(x + 5) (x - 3)$ .

Here is how to find an equivalent expression:

Distribute the $x$ to $(x - 3)$ , and then distribute the $5$ to $(x - 3)$ . Remember, you can use FOIL.

$x^{2} - 3 x + 5 x - 15$

Next, combine like terms.

$x^{2} + 2 x - 15$