SOLUTION: factorise 64x^3+1/27+16x^2+4/3x

Question 353773: factorise 64x^3+1/27+16x^2+4/3x
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Factor (not factorise)
This one is a little tricky. The four terms do not have a Greatest Common Factor (GCF) other than 1. And four terms is too many for factoring by patterns or for trinomial factoring. So we are left with factoring by grouping or factoring by trial and error of the possible rational roots.

When factoring by grouping you usually look for subexpressions which will factor and which, after factoring, have a common factor between them. And many times the factoring of the subexpressions is just simple GCF factoring. But in this problem, the first two terms can be factored using a pattern. Let's see how this works.

First I'll group the two subexpressions which will be factored individually:

Since and , the first subexpression is a sum of cubes which will factor according to the pattern: . In the second subexpression we can factor out a GCF of 4x. Now we have:

which simplifies to:

We can now see that the two factored subexpressions have a common factor: . Factoring this out from both subexpressions we get:

Simplifying the "other" factor:

When factoring, always keep factoring until no more factoring can be done. And the second factor above will factor further. Since and and , the second factor fits the pattern. Using this pattern to factor we get:

which simplifies to:

This problem could be done much faster if...

You know the pattern for (which is not often taught):
You rearrange the terms in order of exponents (highest to lowest):
Recognize that the above expression fits the pattern for with the "a" being 4x and the "b" being 1/3:

If all of the above were true then we could go straight from

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