How do you factor $x^{6} + 16 x^{3} + 64$ $x^6 + 16x^3 + 64$ ?

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Answer 1 · 2016-03-12T19:13:52

We have that

$x^{6} + 16 x^{3} + 64 = {(x^{3})}^{2} + 2 \cdot 8 \cdot x^{3} + {(8)}^{2} = {(x^{3} + 8)}^{2}$ $x^6 + 16x^3 + 64=(x^3)^2+2*8*x^3+(8)^2=(x^3+8)^2$

But $x^{3} + 8$ $x^3+8$ can be factored further as follows

$x^{3} + 8 = x^{3} + 2^{3} = (x + 2) \cdot (x^{2} - 2 x + 4)$ $x^3+8=x^3+2^3=(x+2)*(x^2-2x+4)$

Finally we get

$x^{6} + 16 x^{3} + 64 = {(x + 2)}^{2} \cdot {(x^{2} - 2 x + 4)}^{2}$ $x^6+16x^3+64=(x+2)^2*(x^2-2x+4)^2$

How do you factor x6+16x3+64 x^6 + 16x^3 + 64?