How do you factor x9−x6−x3+1?

2 Answers
Aiima K.
Mar 11, 2018

(x3−1)2(x3+1)

Explanation:

x9−x6−x3−1
x6(x3−1)−1(x3−1)
(x3−1)(x6−1)
(x3−1)[(x3)2−(12)]
(x3−1)(x3−1)(x3+1)
(x3−1)2(x3+1)

Evan
Mar 11, 2018

(x+1)(x−1)2(x2−x+1)(x2+x+1)2

Explanation:

Let's revise our power rules ( they will come in handy later ):

  1. Difference of squares rule: x2−y2=(x−y)(x+y)
  2. Difference of cubes rule: x3−y3=(x−y)(x2+xy+y2)
  3. Sum of cubes rule: x3+y3=(x+y)(x2−xy+y2)

x9−x6−x3+1

Factorise x6 from the first two terms,

x6(x3−1)−x3+1
x6(x3−1)−(x3−1)

Factorise (x3−1),

(x6−1)(x3−1)

Apply difference of squares rule to (x6−1),

(x3+1)(x3−1)(x3−1)

Apply sum of cubes rule to (x3+1),

(x+1)(x2−x+1)(x3−1)(x3−1)

Apply difference of cubes rule to (x3−1),

(x+1)(x2−x+1)(x−1)(x2+x+1)(x−1)(x2+x+1)

Simplify like polynomials,

(x+1)(x−1)2(x2−x+1)(x2+x+1)2

There you go.

P.S. I did not include the sum of squares rule because that rule dwells into imaginary numbers but if you are keen to know, here it is: x2+y2=(x+yi)(x−yi)

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