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Factoring a Sum or Difference of Two Cubes
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Solving Quadratic Equations by Completing the Square
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Factoring a Sum or Difference of Two Cubes

This product will lead us to another factoring pattern.

(a + b)(a2 - ab + b2)

To find the product, multiply each term in (a + b) by each term in (a2 - ab + b2).

To find the product, multiply each term in (a + b) by each term in (a2 - ab + b2).

= a · a2 - a · ab + a · b2 + b · a2 - b · ab + b · b2

= a3 - a2b + ab2 + a2b - ab2 + b3

Combine like terms.

= a3 + b3

The four middle terms add to zero.

The result is a3 + b3, the sum of two cubes.

Note the structure of the two terms:

• The first term, a3, is a perfect cube.

• The last term, b3, is a perfect cube.

• The terms are added.

When we recognize this pattern, we can immediately factor the sum of two cubes as follows:

a3 + b3 = (a + b)(a2 - ab + b2)

A similar factoring pattern holds for a3 - b3, the difference of two cubes.

Note:

You may want to memorize a few perfect cubes.

13 = 1

23 = 8

33 = 27

43 = 64

53 = 125

To check if a number is a perfect cube, use the  key on your calculator to see if the cube root is an integer.

 

Pattern — To Factor the Sum or Difference of Two Cubes

a3 + b3 = (a + b)(a2 - ab + b2)

a3 - b3 = (a - b)(a2 + ab + b2)

 

Example 1

Factor: y3 + 64

Solution

Step 1 Decide if the given polynomial fits a pattern.

The first term, y3, is a perfect cube, (y)3.

The last term, 64, is a perfect cube, (4)3.

The terms are added.

Therefore, y3 + 64 is a sum of two cubes.

Step 2 Identify a and b. Then substitute in the pattern and simplify.
In the factoring pattern for a sum of two cubes, substitute y for a and 4 for b.

 a3 + b3

= (a + b)(a2 - ab + b2)
Simplify.

(y)3 + (4)3

= (y + 4)(y2 - y · 4 + 42)

= (y + 4)(y2 - 4y + 16)

 

The result is:

y3 + 64 = (y + 4)(y2 - 4y + 16).

You can multiply to check the factorization. We leave the check to you.

 

Example 2

Factor: 8w3 - 1

Solution

Step 1 Decide if the given polynomial fits a pattern.

The first term, 8w3, is a perfect cube, (2w)3.

The last term, 1, is a perfect cube, (1)3.

The terms are subtracted.

Therefore, 8w3 - 1 is a difference of two cubes.

Step 2 Identify a and b. Then substitute in the pattern and simplify.

In the factoring pattern for a difference of two cubes, substitute 2w for a and 1 for b.

 

 

Simplify.

 a3 - b3

(2w)3 - (1)3

= (a - b)(a2 + ab + b2)

= (2w - 1)[(2w)2 + 2w · 1 + (1)2]

= (2w - 1)(4w2 + 2w + 1)

The result is: 8w3 - 1 = (2w - 1)(4w2 + 2w + 1).

You can multiply to check the factorization. We leave the check to you.

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