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Algebra 1

7.9.2 Finding Products of Differences

Algebra 17.9.2 Finding Products of Differences

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Activity

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

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

a. ( x + 1 ) ( x 1 ) ( x + 1 ) ( x 1 )

c. ( x 2 ) ( x + 3 ) ( x 2 ) ( x + 3 )

( x 2 ) 2 ( x 2 ) 2

Video: Finding Products of Differences

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

Self Check

Which of the following expressions is equivalent to ( x 2 ) 2 ?
  1. x 2 + 4 x + 4
  2. x 2 4 x + 4
  3. x 2 + 4
  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 and ( 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 ) by ( x 3 ) ( x 3 ) . So, ( x 3 ) 2 = ( x 3 ) ( x 3 ) ( 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 x to the second binomial, then distribute the 3 3 to the second binomial.

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.

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

x 2 + 2 x 4 x 8 = x 2 2 x + 8 x 2 + 2 x 4 x 8 = x 2 2 x + 8

Try it

Try It: Multiplying Binomials with Negative Numbers

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

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