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I'm am doing an 8th grade math text book, and I came across this simple problem:

$8l^3 - 36l^2m + 54lm^2 - 27m^3$ simplifies to?

I immediately got to know that it is $(2l - 3m)^3$ , but how do you Factor it (step by step)? Basically how do you factor expressions which has more than 3 or 4 terms.

Please help.

The variables here and l[l for last] and m. (I don't know how to type symbols, italics { it would be nice if someone includes that also })

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You can look here for a starter on $\LaTeX$, then look on the web for tutorials, then look at what people have done here. – Ross Millikan Jan 6 '14 at 4:47

3 Answers 3

For classwork, there will always be a simple answer. A slightly less "out of the blue" solution is to take the cube roots of the first and last terms, put the proper sign between, and see if it works. Here you win. A second approach is like the rational root theorem-take $l$ as a constant and find the list of roots in $m$. Now for each root $r$, divide by $m-r$ and see if it comes out. You can do it from the other end as well. If it works, you are down to a quadratic, and the quadratic formula or factoring techniques will suffice.

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Hint: substitute $x=2l$ and $y=3m$ into your expression that you want to simplify. Recognize anything yet?

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thank you very much. – Ramana Jan 6 '14 at 4:50
He recognized the factorization. His question was, "Is there a way to do it in general (step-by-step)?" – user44197 Jan 6 '14 at 4:50

As a general rule, you can't really factor it. Imagine if you had a 5th degree polynomial? Galois pretty much proved that there is no general way to factor it (or even extract one factor).

For second degree polynomials, you can always complete the squares.

For third and higher you have to guess by looking at the coefficients. Fixing $m$ and solving for $l$ is too painful!

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