Question

How do you write `y=x^3+6x^2-25x-150` in factored form?

Answer 1

`y = (x-5)(x+5)(x+6)`

Explanation:

The difference of squares identity can be written:

`A^2-B^2 = (A-B)(A+B)`

We will use this with `A=x` and `B=5`.

Given:

`y = x^3+6x^2-25x-150`

Note that the ratio between the first and second terms is the same as that between the third and fourth terms.

So this cubic will factor by grouping:

`y = x^3+6x^2-25x-150`

`color(white)(y) = (x^3+6x^2)-(25x+150)`

`color(white)(y) = x^2(x+6)-25(x+6)`

`color(white)(y) = (x^2-25)(x+6)`

`color(white)(y) = (x^2-5^2)(x+6)`

`color(white)(y) = (x-5)(x+5)(x+6)`