How do you factor the expression $x^{9} - 27 y^{6}$ $x^9-27y^6$ ?

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How do you factor the expression $x^{9} - 27 y^{6}$ $x^9-27y^6$ ?

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Answer 1 · 2018-04-07T14:27:01

$x^{9} - 27 y^{6} = (x^{3} - 3 y^{2}) (x^{6} + 3 x^{3} y^{2} + 9 y^{4})$ $x^9-27y^6 = (x^3-3y^2)(x^6+3x^3y^2+9y^4)$

Explanation:

Given:

$x^{9} - 27 y^{6}$ $x^9-27y^6$

Note that both $x^{9} = {(x^{3})}^{3}$ $x^9 = (x^3)^3$ and $27 y^{6} = {(3 y^{2})}^{3}$ $27y^6 = (3y^2)^3$ are perfect cubes.

So we can factor the given expression as a difference of cubes:

$A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2})$ $A^3-B^3 = (A-B)(A^2+AB+B^2)$

with $A = x^{3}$ $A=x^3$ and $B = 3 y^{2}$ $B=3y^2$ as follows:

$x^{9} - 27 y^{6} = {(x^{3})}^{3} - {(3 y^{2})}^{3}$ $x^9-27y^6 = (x^3)^3-(3y^2)^3$

$x^{9} - 27 y^{6} = (x^{3} - 3 y^{2}) ({(x^{3})}^{2} + (x^{3}) (3 y^{2}) + {(3 y^{2})}^{2})$ $color(white)(x^9-27y^6) = (x^3-3y^2)((x^3)^2+(x^3)(3y^2)+(3y^2)^2)$

$x^{9} - 27 y^{6} = (x^{3} - 3 y^{2}) (x^{6} + 3 x^{3} y^{2} + 9 y^{4})$ $color(white)(x^9-27y^6) = (x^3-3y^2)(x^6+3x^3y^2+9y^4)$

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How do you factor the expression $x^{9} - 27 y^{6}$ $x^9-27y^6$ ?

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How do you factor the expression $x^{9} - 27 y^{6}$ $x^9-27y^6$ ?