Posted by **Z32** on Thursday, September 17, 2009 at 7:56pm.

I wanted to know how you would know that -4x^2 + 2x +90 / x-5 would be able to factor down to (x-5)(-4x-18)/x-5

By looking at -4x^2 + 2x +90 / x-5 , I would never think that it could be factored down to (x-5)(-4x-18)/x-5

How should I approach it so I know that it could be factored down to (x-5)(-4x-18)/x-5?

- Calculus -
**MathMate**, Thursday, September 17, 2009 at 8:18pm
First you take out common factors.

-4x^2 + 2x +90 = -2(2x²-x-45)

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*5-9=1, we can try the factors 5 and 9 in different ways to get:

(2x+9)(x-5)

- Calculus -
**Reiny**, Thursday, September 17, 2009 at 8:20pm
rather hard to explain in detail here.

this page

http://www.recitfga.qc.ca/english/activities/sitsat-2006/jean-foster/0-3.htm

gives a reasonable explanation.

I became suspicious when I saw the simple factor x-5 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 x-a 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 = (x-5)(-4x-18)

BTW, the answer can be taken further by dividing out the x-5 and simplifying the top

(x-5)(-4x-18)/x-5

= -2(2x + 9), x cannot be equal to 5

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