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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 - ,

    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:

  • Calculus - ,

    rather hard to explain in detail here.

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

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