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I am trying to factor the following polynomial: $$ 8x^3 -4x^2y -18xy^2 + 9y^3 $$

$$ (a-b)^3 = a^3 -3a^2b + 3ab^2 - b^3 $$ Thanks

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up vote 4 down vote accepted

Look at the following factorisation, I thought of:


I also want to add that, it is natural to think of the cubic identity you gave us, but $9y^3$ doesn't look good when trying to write as a perfect cube. Also, in particular, in using any identity of this kind, intuitively, since $x^2y$ term has a negative sign, it shoud have come from coefficient of $y$, which means the $y^3$ term must have had a negative coefficient, which is not the case!

Also the negative sign in $xy^2$ suggests on the similar line of thinking that, $x^3$ should have had a negative sign which is also not the case. So, this identity is not worth pursuing here!

Hope this helps!

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Hint: Maybe look at the pretty much equivalent problem of factoring $8x^3-4x^2-18x+9$.

We can use the Rational Roots Theorem to find the rational roots of this, if any (and there are). We can also make life simpler by writing $2x=w$, which yields $w^3-w^2-9w+9$.

Or else we can note that $8x^2-4x^2-18x+9=4x^2(2x^2-1)-9(2x-1)$.

Or else we can start from the original expression, and write it as $4x^2(2x-y)-9(2x-y)$.

And there are other ways.

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