Algebra Examples

Factor x^6-1
x6-1
Step 1
Rewrite x6 as (x2)3.
(x2)3-1
Step 2
Rewrite 1 as 13.
(x2)3-13
Step 3
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=x2 and b=1.
(x2-1)((x2)2+x21+12)
Step 4
Simplify.
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Step 4.1
Rewrite 1 as 12.
(x2-12)((x2)2+x21+12)
Step 4.2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x and b=1.
(x+1)(x-1)((x2)2+x21+12)
Step 4.3
Multiply x2 by 1.
(x+1)(x-1)((x2)2+x2+12)
(x+1)(x-1)((x2)2+x2+12)
Step 5
Multiply the exponents in (x2)2.
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Step 5.1
Apply the power rule and multiply exponents, (am)n=amn.
(x+1)(x-1)(x22+x2+12)
Step 5.2
Multiply 2 by 2.
(x+1)(x-1)(x4+x2+12)
(x+1)(x-1)(x4+x2+12)
Step 6
One to any power is one.
(x+1)(x-1)(x4+x2+1)
Step 7
Factor.
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Step 7.1
Rewrite x4+x2+1 in a factored form.
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Step 7.1.1
Rewrite the middle term.
(x+1)(x-1)(x4+2x21-x2+1)
Step 7.1.2
Rearrange terms.
(x+1)(x-1)(x4+2x21+1-x2)
Step 7.1.3
Factor first three terms by perfect square rule.
(x+1)(x-1)((x2+1)2-x2)
Step 7.1.4
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x2+1 and b=x.
(x+1)(x-1)((x2+1+x)(x2+1-x))
Step 7.1.5
Simplify.
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Step 7.1.5.1
Reorder terms.
(x+1)(x-1)((x2+x+1)(x2+1-x))
Step 7.1.5.2
Reorder terms.
(x+1)(x-1)((x2+x+1)(x2-x+1))
(x+1)(x-1)((x2+x+1)(x2-x+1))
(x+1)(x-1)((x2+x+1)(x2-x+1))
Step 7.2
Remove unnecessary parentheses.
(x+1)(x-1)(x2+x+1)(x2-x+1)
(x+1)(x-1)(x2+x+1)(x2-x+1)
x6-1
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