Question

How do you factor `4(x^3)-8(x^2)-25x+50`?

Answer 1

`4x^3-8x^2-25x+50=(2x-5)(2x+5)(x-2)`

Explanation:

The difference of squares identity can be written:

`a^2-b^2=(a-b)(a+b)`

We use this with `a=2x` and `b=5` later...

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Note that the ratio of the first and second terms is the same as that between the third and fourth terms. So this cubic will factor by grouping:

`4x^3-8x^2-25x+50`

`=(4x^3-8x^2)-(25x-50)`

`=4x^2(x-2)-25(x-2)`

`=(4x^2-25)(x-2)`

`=((2x)^2-5^2)(x-2)`

`=(2x-5)(2x+5)(x-2)`