Step by step solution to "How do you factor x3−27 ?"

Katherine Walls

Answered question

2021-12-22

How do you factor

x^{3} - 27

twineg4

Beginner2021-12-23Added 33 answers

Both

x^{3}

and

27 = 3^{3}

are perfect cubes. So we can use the difference of cubes identity:

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

with

a = x

b = 3

as follows:

x - 27 = x^{3} - 3^{3}

= (x - 3) (x^{2} + x (3) + 3^{2})

= (x - 3) (x^{2} + 3 x + 9)

This is as far as you can go with Real coefficients. If you allow Complex coefficients then you can factor this a little further:

= (x - 3) (x - 3 ω) (x - 3 ω^{2})

where

ω = - \frac{12}{+} \frac{\sqrt{3}}{2} i

is the primitive Complex cube root of 1

alexandrebaud43

Beginner2021-12-24Added 36 answers

Rewrite 27 as

3^{3}

x^{3} - 3^{3}

Since both terms are perfect cubes, factor using the difference of cubes formula,

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

where

a = x

and

b = 3

(x - 3) (x^{2} + x \cdot 3 + 3^{2})

Simplify.
Move 3 to the left of x.

(x - 3) (x^{2} + 3 x + 3^{2})

Raise 3 to the power of 2.

(x - 3) (x^{2} + 3 x + 9)

karton

Expert2021-12-30Added 613 answers

Very detailed answer. Thanks.

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