Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

2 Answers 2

up vote 4 down vote accepted

Look at the following factorisation, I thought of:

$$\begin{align*}8x^3-4x^2y-18xy^2+9y^3&=4x^2(2x-y)-9y^2(2x-y)\\&=(4x^2-9y^2)(2x-y)\\&=(2x+3y)(2x-3y)(2x-y)\end{align*}$$

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!

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.