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6.4 Special products  (Page 3/5)

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Multiply: ( x − 5 ) ( x + 5 ) .

x 2 − 25

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Multiply: ( w − 3 ) ( w + 3 ) .

w 2 − 9

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Multiply: ( 2 x + 5 ) ( 2 x − 5 ) .

Solution

Are the binomials conjugates?

It is the product of conjugates. The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.
Square the first term, 2 x . 2 x squared minus blank. Above this is the general form a squared minus b squared.
Square the last term, 5. 2 x squared minus 5 squared.
Simplify. The product is a difference of squares. 4 x squared minus 25.

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Multiply: ( 6 x + 5 ) ( 6 x − 5 ) .

36 x 2 − 25

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Multiply: ( 2 x + 7 ) ( 2 x − 7 ) .

4 x 2 − 49

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The binomials in the next example may look backwards – the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.

Find the product: ( 3 + 5 x ) ( 3 − 5 x ) .

Solution

It is the product of conjugates. The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.
Use the pattern. 3 squared minus 5 x squared. Above this is the general form a squared minus b squared.
Simplify. 9 minus 25 x squared.

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Multiply: ( 7 + 4 x ) ( 7 − 4 x ) .

49 − 16 x 2

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Multiply: ( 9 − 2 y ) ( 9 + 2 y ) .

81 − 4 y 2

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Now we’ll multiply conjugates that have two variables.

Find the product: ( 5 m − 9 n ) ( 5 m + 9 n ) .

Solution

This fits the pattern. 5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.
Use the pattern. 5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.
Simplify. 25 m squared minus 81 n squared.

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Find the product: ( 4 p − 7 q ) ( 4 p + 7 q ) .

16 p 2 − 49 q 2

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Find the product: ( 3 x − y ) ( 3 x + y ) .

9 x 2 − y 2

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Find the product: ( c d − 8 ) ( c d + 8 ) .

Solution

This fits the pattern. The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.
Use the pattern. c d squared minus 8 squared. Above this is the general form a squared minus b squared.
Simplify. c squared d squared minus 64.

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Find the product: ( x y − 6 ) ( x y + 6 ) .

x 2 y 2 − 36

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Find the product: ( a b − 9 ) ( a b + 9 ) .

a 2 b 2 − 81

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Find the product: ( 6 u 2 − 11 v 5 ) ( 6 u 2 + 11 v 5 ) .

Solution

This fits the pattern. The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.
Use the pattern. 6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.
Simplify. 36 u to the fourth power minus 121 v to the tenth power.

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Find the product: ( 3 x 2 − 4 y 3 ) ( 3 x 2 + 4 y 3 ) .

9 x 4 − 16 y 6

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Find the product: ( 2 m 2 − 5 n 3 ) ( 2 m 2 + 5 n 3 ) .

4 m 4 − 25 n 6

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Recognize and use the appropriate special product pattern

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

Comparing the special product patterns

Binomial Squares Product of Conjugates ( a + b ) 2 = a 2 + 2 a b + b 2 ( a − b ) ( a + b ) = a 2 − b 2 ( a − b ) 2 = a 2 − 2 a b + b 2 - Squaring a binomial - Multiplying conjugates - Product is a trinomial - Product is a binomial - Inner and outer terms with FOIL are the same. - Inner and outer terms with FOIL are opposites. - Middle term is double the product of the terms. - There is no middle term .

Choose the appropriate pattern and use it to find the product:

ⓐ ( 2 x − 3 ) ( 2 x + 3 ) ⓑ ( 5 x − 8 ) 2 ⓒ ( 6 m + 7 ) 2 ⓓ ( 5 x − 6 ) ( 6 x + 5 )

Solution

  1. ⓐ ( 2 x − 3 ) ( 2 x + 3 ) These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.
    This fits the pattern. The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.
    Use the pattern. 2 x squared minus 3 squared. Above this is the general form a squared minus b squared.
    Simplify. 4 x squared minus 9.
  2. ⓑ ( 8 x − 5 ) 2 We are asked to square a binomial. It fits the binomial squares pattern.
    8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.
    Use the pattern. 8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.
    Simplify. 64 x squared minus 80 x plus 25.
  3. ⓒ ( 6 m + 7 ) 2 Again, we will square a binomial so we use the binomial squares pattern.
    6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.
    Use the pattern. 6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.
    Simplify. 36 m squared plus 84 m plus 49.
  4. ⓓ ( 5 x − 6 ) ( 6 x + 5 ) This product does not fit the patterns, so we will use FOIL.
    ( 5 x − 6 ) ( 6 x + 5 ) Use FOIL. 30 x 2 + 25 x − 36 x − 30 Simplify. 30 x 2 − 11 x − 30
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Choose the appropriate pattern and use it to find the product:

ⓐ ( 9 b − 2 ) ( 2 b + 9 ) ⓑ ( 9 p − 4 ) 2 ⓒ ( 7 y + 1 ) 2 ⓓ ( 4 r − 3 ) ( 4 r + 3 )

ⓐ FOIL; 18 b 2 + 77 b − 18 ⓑ Binomial Squares; 81 p 2 − 72 p + 16 ⓒ Binomial Squares; 49 y 2 + 14 y + 1 ⓓ Product of Conjugates; 16 r 2 − 9

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Choose the appropriate pattern and use it to find the product:

ⓐ ( 6 x + 7 ) 2 ⓑ ( 3 x − 4 ) ( 3 x + 4 ) ⓒ ( 2 x − 5 ) ( 5 x − 2 ) ⓓ ( 6 n − 1 ) 2

ⓐ Binomial Squares; 36 x 2 + 84 x + 49 ⓑ Product of Conjugates; 9 x 2 − 16 ⓒ FOIL; 10 x 2 − 29 x + 10 ⓓ Binomial Squares; 36 n 2 − 12 n + 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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