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Recognizing and Using the Appropriate Special Product Pattern

Recognizing and Using 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

\(\begin{array}{cccc}\mathbf{\text{Binomial Squares}}\hfill & & & \mathbf{\text{Product of Conjugates}}\hfill \\ {\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\hfill & & & \left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}\hfill \\ {\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\hfill & & & \\ \text{- Squaring a binomial}\hfill & & & \text{- Multiplying conjugates}\hfill \\ \text{- Product is a}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{trinomial}}\hfill & & & \text{- Product is a}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{binomial}}\hfill \\ \text{- Inner and outer terms with FOIL are}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{the same.}}\hfill & & & \text{- Inner and outer terms with FOIL are}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{opposites.}}\hfill \\ \text{- Middle term is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{double the product}}\phantom{\rule{0.2em}{0ex}}\text{of the terms.}\hfill & & & \text{- There is}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{no}}\phantom{\rule{0.2em}{0ex}}\text{middle term}.\hfill \end{array}\)

Example

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

  1. \(\left(2x-3\right)\left(2x+3\right)\)
  2. \({\left(5x-8\right)}^{2}\)
  3. \({\left(6m+7\right)}^{2}\)
  4. \(\left(5x-6\right)\left(6x+5\right)\)

Solution

  1. \(\left(2x-3\right)\left(2x+3\right)\) 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. \({\left(8x-5\right)}^{2}\) 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. \({\left(6m+7\right)}^{2}\) 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. \(\left(5x-6\right)\left(6x+5\right)\) This product does not fit the patterns, so we will use FOIL.\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\left(5x-6\right)\left(6x+5\right)\hfill \\ \text{Use FOIL.}\hfill & & & \phantom{\rule{4em}{0ex}}30{x}^{2}+25x-36x-30\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}30{x}^{2}-11x-30\hfill \end{array}\)

Optional Video:

You could watch the video below for additional instruction and practice with special products:

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