Summarizing Factor Special Products

Key Concepts

• Factor perfect square trinomials
  \(\begin{array}{ccccccc}\text{Step 1.}\phantom{\rule{0.2em}{0ex}}\text{Does the trinomial fit the pattern?}\hfill & & & \hfill {a}^{2}+2ab+{b}^{2}\hfill & & & \hfill {a}^{2}-2ab+{b}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Is the first term a perfect square?}\hfill & & & \hfill {(a)}^{2}\hfill & & & \hfill {(a)}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}\text{Is the last term a perfect square?}\hfill & & & \hfill {(a)}^{2}\phantom{\rule{4.5em}{0ex}}{(b)}^{2}\hfill & & & \hfill {(a)}^{2}\phantom{\rule{4.5em}{0ex}}{(b)}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}\text{Check the middle term. Is it}\phantom{\rule{0.2em}{0ex}}2ab?\hfill & & & \hfill {(a)}^{2}{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{(b)}^{2}\hfill & & & \hfill {(a)}^{2}{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{(b)}^{2}\hfill \\ \text{Step 2.}\phantom{\rule{0.2em}{0ex}}\text{Write the square of the binomial.}\hfill & & & \hfill {(a+b)}^{2}\hfill & & & \hfill {(a-b)}^{2}\hfill \\ \text{Step 3.}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & & & & \end{array}\)
• Factor differences of squares
  \(\begin{array}{cccc}\text{Step 1.}\phantom{\rule{0.2em}{0ex}}\text{Does the binomial fit the pattern?}\hfill & & & \hfill {a}^{2}-{b}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Is this a difference?}\hfill & & & \hfill \_\_\_\_-\_\_\_\_\hfill \\ \phantom{\rule{3em}{0ex}}\text{Are the first and last terms perfect squares?}\hfill & & & \\ \text{Step 2.}\phantom{\rule{0.2em}{0ex}}\text{Write them as squares.}\hfill & & & \hfill {(a)}^{2}-{(b)}^{2}\hfill \\ \text{Step 3.}\phantom{\rule{0.2em}{0ex}}\text{Write the product of conjugates.}\hfill & & & \hfill (a-b)(a+b)\hfill \\ \text{Step 4.}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & \end{array}\)
• Factor sum and difference of cubes To factor the sum or difference of cubes:
  1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes?
  2. Write them as cubes.
  3. Use either the sum or difference of cubes pattern.
  4. Simplify inside the parentheses
  5. Check by multiplying the factors.

Glossary

perfect square trinomials pattern

If a and b are real numbers,

difference of squares pattern

If a and b are real numbers,

sum and difference of cubes pattern

\(\begin{array}{c}\hfill {a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})\hfill \\ \hfill {a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\hfill \end{array}\)