Steply

Sum and difference of cubes

Algebra 1 · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Common factor first
  2. Write as cubes
  3. Use the pattern
  4. Tidy up
  5. Check

The steps for Factor 2x^3 + 250. Every solution names its steps, so you always know what comes next.

Worked examples

Factor x^3 - 8 Easy

The question

Question: factor x³ − 8.

x3 − 8

1 Write as cubes

Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.

x³ is x cubed.

8 is 2 cubed.

cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
x3 = (x)3, 8 = 2³

So it is a difference of two cubes: a is x and b is 2.

= (x)3 − 23

2 Use the pattern

Here is the pattern.

The first bracket has the same sign as the question.

In the second bracket the middle sign is the opposite one, and the last sign is always plus.

a³ − b³ = (a − b)(a² + ab + b²)
first bracket: same sign (−); middle: opposite (+); last: +

We put a = x and b = 2 into the pattern.

= (x − 2)(x2 + x × 2 + 22)

Careful!

It is not (x − 2)³.

Multiplied out, that has x² and x terms too.

Careful: not (x − 2)³ ✗

3 Tidy up

x squared is x².

x times 2 is 2x.

2 squared is 4.

(x)² = x², x × 2 = 2x, 2² = 4

So we get (x − 2)(x² + 2x + 4).

= (x − 2)(x2 + 2x + 4)

The second bracket can't be factored: with this pattern it never can.

4 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (x − 2)(x2 + 2x + 4)
= x3 + 2x2 + 4x − 2x2 − 4x − 8 = x3 − 8

We get x³ − 8 again.

So it is right!

Let's look back.

First: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.

Answer
= (x − 2)(x2 + 2x + 4)

▶ Watch it solved

Factor 8x^3 - 1 Medium

The question

Question: factor 8x³ − 1.

8x3 − 1

1 Write as cubes

Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.

8x³ is 2x cubed.

1 is 1 cubed: 1 × 1 × 1 = 1.

cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
8x3 = (2x)3, 1 = 1³

So it is a difference of two cubes: a is 2x and b is 1.

= (2x)3 − 13

2 Use the pattern

Here is the pattern.

The first bracket has the same sign as the question.

In the second bracket the middle sign is the opposite one, and the last sign is always plus.

a³ − b³ = (a − b)(a² + ab + b²)
first bracket: same sign (−); middle: opposite (+); last: +

We put a = 2x and b = 1 into the pattern.

= (2x − 1)((2x)2 + 2x × 1 + 12)

Careful!

It is not (2x − 1)³.

Multiplied out, that has x² and x terms too.

Careful: not (2x − 1)³ ✗

3 Tidy up

2x squared is 4x².

2x times 1 is 2x.

1 squared is 1.

(2x)² = 4x², 2x × 1 = 2x, 1² = 1

So we get (2x − 1)(4x² + 2x + 1).

= (2x − 1)(4x2 + 2x + 1)

The second bracket can't be factored: with this pattern it never can.

4 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (2x − 1)(4x2 + 2x + 1)
= 8x3 + 4x2 + 2x − 4x2 − 2x − 1 = 8x3 − 1

We get 8x³ − 1 again.

So it is right!

Let's look back.

First: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.

Answer
= (2x − 1)(4x2 + 2x + 1)

▶ Watch it solved

Factor 2x^3 + 250 Harder

The question

Question: factor 2x³ + 250.

2x3 + 250

1 Common factor first

Always look for a common factor first.

2 and 250 can both be divided by 2.

2, 250: all in the 2 times table

So it is 2(x³ + 125).

= 2(x3 + 125)

2 Write as cubes

Cubes are a number times itself three times: 2 × 2 × 2 = 8, 3 × 3 × 3 = 27.

x³ is x cubed.

125 is 5 cubed.

cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
x3 = (x)3, 125 = 5³

So it is a sum of two cubes: a is x and b is 5.

= 2((x)3 + 53)

3 Use the pattern

Here is the pattern.

The first bracket has the same sign as the question.

In the second bracket the middle sign is the opposite one, and the last sign is always plus.

a³ + b³ = (a + b)(a² − ab + b²)
first bracket: same sign (+); middle: opposite (−); last: +

We put a = x and b = 5 into the pattern.

= 2[(x + 5)(x2 − x × 5 + 52)]

Careful!

It is not (x + 5)³.

Multiplied out, that has x² and x terms too.

Careful: not (x + 5)³ ✗

4 Tidy up

x squared is x².

x times 5 is 5x.

5 squared is 25.

(x)² = x², x × 5 = 5x, 5² = 25

So we get 2(x + 5)(x² − 5x + 25).

= 2(x + 5)(x2 − 5x + 25)

The second bracket can't be factored: with this pattern it never can.

5 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (x + 5)(x2 − 5x + 25)
= x3 − 5x2 + 25x + 5x2 − 25x + 125
= x3 + 125

Then the common factor times every part.

2(x3 + 125) = 2x3 + 250

We get 2x³ + 250 again.

So it is right!

Let's look back.

First: common factor first.

Then: write as cubes.

Then: use the pattern.

Then: tidy up.

Last: check.

Answer
= 2(x + 5)(x2 − 5x + 25)

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. x^3 - 8
    Answer
    = (x − 2)(x2 + 2x + 4)
  2. x^3 + 27
    Answer
    = (x + 3)(x2 − 3x + 9)
  3. x^3 - 1
    Answer
    = (x − 1)(x2 + x + 1)
  4. y^3 + 64
    Answer
    = (y + 4)(y2 − 4y + 16)
  5. x^3 - 125
    Answer
    = (x − 5)(x2 + 5x + 25)
  6. a^3 + 1
    Answer
    = (a + 1)(a2 − a + 1)
  7. x^3 - 216
    Answer
    = (x − 6)(x2 + 6x + 36)
  8. x^3 + 1000
    Answer
    = (x + 10)(x2 − 10x + 100)

Medium

  1. 8x^3 - 1
    Answer
    = (2x − 1)(4x2 + 2x + 1)
  2. 27x^3 + 8
    Answer
    = (3x + 2)(9x2 − 6x + 4)
  3. 64y^3 - 27
    Answer
    = (4y − 3)(16y2 + 12y + 9)
  4. 8x^3 + 125
    Answer
    = (2x + 5)(4x2 − 10x + 25)
  5. 125x^3 - 1
    Answer
    = (5x − 1)(25x2 + 5x + 1)
  6. 2x^3 - 54
    Answer
    = 2(x − 3)(x2 + 3x + 9)
  7. 3x^3 + 24
    Answer
    = 3(x + 2)(x2 − 2x + 4)
  8. 5a^3 - 40
    Answer
    = 5(a − 2)(a2 + 2a + 4)

Harder

  1. 2x^3 + 250
    Answer
    = 2(x + 5)(x2 − 5x + 25)
  2. 10x^3 + 10
    Answer
    = 10(x + 1)(x2 − x + 1)
  3. 16x^3 - 54
    Answer
    = 2(2x − 3)(4x2 + 6x + 9)
  4. 24x^3 + 81
    Answer
    = 3(2x + 3)(4x2 − 6x + 9)
  5. 8 - x^3
    Answer
    = −(x − 2)(x2 + 2x + 4)