Question

How do you factor given that `f(6)=0` and `f(x)=x^3-3x^2-16x-12`?

Answer 1

`f(x)=(x-6)(x+1)(x+2)`

Explanation:

`"using the "color(blue)"factor theorem"`

`•color(white)(x)f(a)=0" if and only if "(x-a)" is a factor of "f(x)`

`f(6)=0rArr(x-6)" is a factor of "f(x)`

`f(x)=color(red)(x^2)(x-6)color(magenta)(+6x^2)-3x^2-16x-12`

`color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(magenta)(+18x)-16x-12`

`color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(red)(+2)(x-6)color(magenta)(+12)-12`

`color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(red)(+2)(x-6)+0`

`rArrf(x)=(x-6)(color(red)(x^2+3x+2))`

`color(white)(rArrf(x))=(x-6)(x+1)(x+2)`