How do you factor $x^{3} - 24 x^{2} + 192 x - 512$ $x^3 - 24x^2 + 192x - 512$ ?

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How do you factor $x^{3} - 24 x^{2} + 192 x - 512$ $x^3 - 24x^2 + 192x - 512$ ?

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Answer 1 · 2015-10-04T10:44:17

Notice that the leading and trailing terms are both perfect cubes, so try and find that:

$x^{3} - 24 x^{2} + 192 x - 512 = {(x - 8)}^{3}$ $x^3-24x^2+192x-512 = (x-8)^3$

Explanation:

In the general case ${(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$

Notice that $x^{3}$ $x^3$ is a perfect cube and $- 512 = {(- 8)}^{3}$ $-512 = (-8)^3$ is also a perfect cube.

So try $a = x$ $a = x$ and $b = - 8$ $b = -8$ to find:

${(x - 8)}^{3} = x^{3} + 3 x^{2} (- 8) + 3 x {(- 8)}^{2} + {(- 8)}^{3}$ $(x-8)^3 = x^3 + 3x^2(-8) + 3x(-8)^2 + (-8)^3$

$= x^{3} - 24 x^{2} + 192 x - 512$ $=x^3-24x^2+192x-512$

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How do you factor $x^{3} - 24 x^{2} + 192 x - 512$ $x^3 - 24x^2 + 192x - 512$ ?

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How do you factor $x^{3} - 24 x^{2} + 192 x - 512$ $x^3 - 24x^2 + 192x - 512$ ?