How would the expression x3 - 8 be rewritten using difference of cubes?

Algebra
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How would the expression x3 - 8 be rewritten using difference of cubes?

Jun 3rd, 2015

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Use the formula of the differenceof cubes which is:   a^3 - b^3 = (a - b)(a^2 + ab + b^2).

So we compare the given expression x^3 - 8 to the left side of the formula: a^3 - b^3.

Here, we can see: a^3 = x^3  and  b^3 = 8. So we solve each equation by taking the cube root like this:

3th root(a^3) = 3th root(x^3) ---------> a = x

3th root(b^3) = 3th root(8) ------------> b = 2  (the cubic root of 8 is 2 since 2*2*2 = 8).

Then we enter x in place of a and 2 in place of b into the right side of the formula like this:

(a - b)(a^2 + ab + b^2) = (x - 2)(x^2 + 2x + 2^2)  = (x - 2)(x^2 + 2x + 4)   since 2^2 = 2*2 = 4.

So x^3 - 8 = (x - 2)(x^2 + 2x + 4).

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 3rd, 2015

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