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Sum and difference of cubes pattern

a 3 + b 3 = ( a + b ) ( a 2 a b + b 2 ) a 3 b 3 = ( a b ) ( a 2 + a b + b 2 )

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.

This figure demonstrates the sign patterns in the sum and difference of two cubes. For the sum of two cubes, this figure shows the first two signs are plus and the first and the third signs are opposite, plus minus. The difference of two cubes has the first two signs the same, minus. The first and the third sign are minus plus.

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] .

This table has two rows. The first row is labeled n. The second row is labeled n cubed. The first row has the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The second row has the perfect cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

How to factor the sum or difference of cubes

Factor: x 3 + 64 .

Solution

This table gives the steps for factoring x cubed + 64. The first step is to verify the binomial fits the pattern. Also, to check the sign for a sum or difference. This binomial is a sum that fits the pattern. The second step is to write the terms as cubes, x cubed + 4 cubed. The third step is follow the pattern for the sum of two cubes, (x + 4)(x squared minus x times 4 + 4 squared). The fourth step is to simplify, (x + 4)(x squared minus 4 x +16). The last step is to check the answer with multiplication.
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Factor: x 3 + 27 .

( x + 3 ) ( x 2 3 x + 9 )

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Factor: y 3 + 8 .

( y + 2 ) ( y 2 2 y + 4 )

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Factor the sum or 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.

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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Factor: u 3 125 .

( u 5 ) ( u 2 + 5 u + 25 )

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Factor: v 3 343 .

( v 7 ) ( v 2 + 7 v + 49 )

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

Factor: 512 125 p 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: 64 27 x 3 .

( 4 3 x ) ( 16 12 x + 9 x 2 )

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Factor: 27 8 y 3 .

( 3 2 y ) ( 9 6 y + 4 y 2 )

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Factor: 27 u 3 125 v 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: 8 x 3 27 y 3 .

( 2 x 3 y ) ( 4 x 2 6 x y + 9 y 2 )

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Factor: 1000 m 3 125 n 3 .

( 10 m 5 n ) ( 100 m 2 50 m n + 25 n 2 )

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

Factor: 5 m 3 + 40 n 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 ( m + 2 n ) ( m 2 2 m n + 4 n 2 )

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Factor: 500 p 3 + 4 q 3 .

4 ( 5 p + q ) ( 25 p 2 5 p q + q 2 )

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Factor: 432 c 3 + 686 d 3 .

2 ( 6 c + 7 d ) ( 36 c 2 42 c d + 49 d 2 )

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Access these online resources for additional instruction and practice with factoring special products.

Key concepts

  • Factor perfect square trinomials See [link] .
    Step 1. Does the trinomial fit the pattern? a 2 + 2 a b + b 2 a 2 2 a b + b 2 Is the first term a perfect square? ( a ) 2 ( a ) 2 Write it as a square. Is the last term a perfect square? ( a ) 2 ( b ) 2 ( a ) 2 ( b ) 2 Write it as a square. Check the middle term. Is it 2 a b ? ( a ) 2 2 · a · b ( b ) 2 ( a ) 2 2 · a · b ( b ) 2 Step 2. Write the square of the binomial. ( a + b ) 2 ( a b ) 2 Step 3. Check by multiplying.
  • Factor differences of squares See [link] .
    Step 1. Does the binomial fit the pattern? a 2 b 2 Is this a difference? ____ ____ Are the first and last terms perfect squares? Step 2. Write them as squares. ( a ) 2 ( b ) 2 Step 3. Write the product of conjugates. ( a b ) ( a + b ) Step 4. Check by multiplying.
  • 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.

16 y 2 + 24 y + 9

( 4 y + 3 ) 2

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36 s 2 + 84 s + 49

( 6 s + 7 ) 2

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100 x 2 20 x + 1

( 10 x 1 ) 2

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25 n 2 120 n + 144

( 5 n 12 ) 2

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49 x 2 28 x y + 4 y 2

( 7 x 2 y ) 2

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25 r 2 60 r s + 36 s 2

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25 n 2 + 25 n + 4

( 5 n + 4 ) ( 5 n + 1 )

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64 m 2 34 m + 1

( 32 m 1 ) ( 2 m 1 )

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10 k 2 + 80 k + 160

10 ( k + 4 ) 2

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75 u 3 30 u 2 v + 3 u v 2

3 u ( 5 u v ) 2

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90 p 3 + 300 p 2 q + 250 p q 2

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Factor Differences of Squares

In the following exercises, factor.

x 2 16

( x 4 ) ( x + 4 )

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25 v 2 1

( 5 v 1 ) ( 5 v + 1 )

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121 x 2 144 y 2

( 11 x 12 y ) ( 11 x + 12 y )

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169 c 2 36 d 2

( 13 c 6 d ) ( 13 c + 6 d )

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4 49 x 2

( 7 x 2 ) ( 7 x + 2 ) ( 2 7 x ) ( 2 + 7 x )

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16 z 4 1

( 2 z 1 ) ( 2 z + 1 ) ( 4 z 2 + 1 )

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5 q 2 45

5 ( q 3 ) ( q + 3 )

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24 p 2 + 54

6 ( 4 p 2 + 9 )

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Factor Sums and Differences of Cubes

In the following exercises, factor.

x 3 + 125

( x + 5 ) ( x 2 5 x + 25 )

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z 3 27

( z 3 ) ( z 2 + 3 z + 9 )

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8 343 t 3

( 2 7 t ) ( 4 + 14 t + 49 t 2 )

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8 y 3 125 z 3

( 2 y 5 z ) ( 4 y 2 + 10 y z + 25 z 2 )

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7 k 3 + 56

7 ( k + 2 ) ( k 2 2 k + 4 )

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2 16 y 3

2 ( 1 2 y ) ( 1 + 2 y + 4 y 2 )

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Mixed Practice

In the following exercises, factor.

64 a 2 25

( 8 a 5 ) ( 8 a + 5 )

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27 q 2 3

3 ( 3 q 1 ) ( 3 q + 1 )

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16 x 2 72 x + 81

( 4 x 9 ) 2

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8 p 2 + 2

2 ( 4 p 2 + 1 )

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125 8 y 3

( 5 2 y ) ( 25 + 10 y + 4 y 2 )

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45 n 2 + 60 n + 20

5 ( 3 n + 2 ) 2

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Everyday math

Landscaping Sue and Alan are planning to put a 15 foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is w , the total area of the pool and deck is given by the trinomial 4 w 2 + 60 w + 225 . Factor the trinomial.

( 2 w + 15 ) 2

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Home repair The height a twelve foot ladder can reach up the side of a building if the ladder’s base is b feet from the building is the square root of the binomial 144 b 2 . Factor the binomial.

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Writing exercises

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

Answers may vary.

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How do you recognize the binomial squares pattern?

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Explain why n 2 + 25 ( n + 5 ) 2 . Use algebra, words, or pictures.

Answers may vary.

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Maribel factored y 2 30 y + 81 as ( y 9 ) 2 . Was she right or wrong? How do you know?

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Self check

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

This table has the following statements all to be preceded by “I can…”. The first row is “factor perfect square trinomials”. The second row is “factor differences of squares”. The third row is “factor sums and differences of cubes”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

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?

Practice Key Terms 3

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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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