Calculus
posted by Z32 .
I wanted to know how you would know that 4x^2 + 2x +90 / x5 would be able to factor down to (x5)(4x18)/x5
By looking at 4x^2 + 2x +90 / x5 , I would never think that it could be factored down to (x5)(4x18)/x5
How should I approach it so I know that it could be factored down to (x5)(4x18)/x5?

First you take out common factors.
4x^2 + 2x +90 = 2(2x²x45)
Is it possible to find a, b such that ab=45, 2a+b=1 or a+2b=1?
Look at the factors of the constant term, namely 45=5*9. Since 2*59=1, we can try the factors 5 and 9 in different ways to get:
(2x+9)(x5) 
rather hard to explain in detail here.
this page
http://www.recitfga.qc.ca/english/activities/sitsat2006/jeanfoster/03.htm
gives a reasonable explanation.
I became suspicious when I saw the simple factor x5 in the denominator, and had a sneaking suspicion that it could also be in the top.
so I subbed in x=5 into the top and sure enough, I got a zero for the result.
(recall the factor theorem, which says that if f(a) = 0 then xa is a factor)
so I know that
4x^2 + 2x +90
= (x  5)(?x ± ?)
now a bit of logic,
what multiplied by x would give me 4x^2 ? , clearly 4x
what multiplied by 5 would give me +90 ? clearly 18
so
4x^2 + 2x +90 = (x5)(4x18)
BTW, the answer can be taken further by dividing out the x5 and simplifying the top
(x5)(4x18)/x5
= 2(2x + 9), x cannot be equal to 5
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