7.4 Factor special products  (Page 4/7)

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.

The trinomial factor in the sum and difference of cubes pattern    cannot be factored.

It can be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in [link] .

How to factor the sum or difference of cubes

Factor: $x^3+64$ .

Solution

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Factor: $x^3-1000$ .

Solution

This binomial is a difference. The first and last terms are perfect cubes.
Write the terms as cubes.
Use the difference of cubes pattern.
Simplify.
Check by multiplying.

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Be careful to use the correct signs in the factors of the sum and difference of cubes.

Factor: $512-125p^3$ .

Solution

This binomial is a difference. The first and last terms are perfect cubes.
Write the terms as cubes.
Use the difference of cubes pattern.
Simplify.
Check by multiplying.	We'll leave the check to you.

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Factor: $27u^3-125v^3$ .

Solution

This binomial is a difference. The first and last terms are perfect cubes.
Write the terms as cubes.
Use the difference of cubes pattern.
Simplify.
Check by multiplying.	We'll leave the check to you.

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In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.

Factor: $5m^3+40n^3$ .

Solution

Factor the common factor.
This binomial is a sum. The first and last terms are perfect cubes.
Write the terms as cubes.
Use the sum of cubes pattern.
Simplify.

Check. To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by 5. We’ll leave the multiplication for you.

$5\left(\stackrel{}{m+2n}\right)\left(\stackrel{}{m^2-2mn+4n^2}\right)$

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Key concepts

• Factor perfect square trinomials See [link] .
  $\begin{array}{ccccccc}\text{Step 1.}\text{Does the trinomial fit the pattern?}\hfill & & & \hfill a^2+2ab+b^2\hfill & & & \hfill a^2-2ab+b^2\hfill \\ \text{Is the first term a perfect square?}\hfill & & & \hfill {\left(a\right)}^2\hfill & & & \hfill {\left(a\right)}^2\hfill \\ \text{Write it as a square.}\hfill & & & & & & \\ \text{Is the last term a perfect square?}\hfill & & & \hfill {\left(a\right)}^2{\left(b\right)}^2\hfill & & & \hfill {\left(a\right)}^2{\left(b\right)}^2\hfill \\ \text{Write it as a square.}\hfill & & & & & & \\ \text{Check the middle term. Is it}2ab?\hfill & & & \hfill {\left(a\right)}^2{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{\left(b\right)}^2\hfill & & & \hfill {\left(a\right)}^2{}_{\text{↘}}\underset{2·a·b}{}{}_{\text{↙}}{\left(b\right)}^2\hfill \\ \text{Step 2.}\text{Write the square of the binomial.}\hfill & & & \hfill {\left(a+b\right)}^2\hfill & & & \hfill {\left(a-b\right)}^2\hfill \\ \text{Step 3.}\text{Check by multiplying.}\hfill & & & & & & \end{array}$
• Factor differences of squares See [link] .
  $\begin{array}{cccc}\text{Step 1.}\text{Does the binomial fit the pattern?}\hfill & & & \hfill a^2-b^2\hfill \\ \text{Is this a difference?}\hfill & & & \hfill \_\_\_\_-\_\_\_\_\hfill \\ \text{Are the first and last terms perfect squares?}\hfill & & & \\ \text{Step 2.}\text{Write them as squares.}\hfill & & & \hfill {\left(a\right)}^2-{\left(b\right)}^2\hfill \\ \text{Step 3.}\text{Write the product of conjugates.}\hfill & & & \hfill \left(a-b\right)\left(a+b\right)\hfill \\ \text{Step 4.}\text{Check by multiplying.}\hfill & & & \end{array}$
• Factor sum and difference of cubes To factor the sum or difference of cubes: See [link] .
  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.

Practice makes perfect

Factor Perfect Square Trinomials

In the following exercises, factor.

Factor Differences of Squares

In the following exercises, factor.

Factor Sums and Differences of Cubes

In the following exercises, factor.

Mixed Practice

In the following exercises, factor.

Everyday math

Writing exercises

Self check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?