Question

Given polynomial `f(x)=x^3-10x^2+19x+30` and a factor `x-6` how do you find all other factors?

Answer 1

`:.x^3-10x^2+19x+30=(x-6)(x-5)(x+1)`

Explanation:

if `(x-6)` is a factor we have

`x^3-10x^2+19x+30=(x-6)(x^2+bx+c)`

`color(white)(xxxxxxxxxxxxxxx)=x^3+bx^2+cx`

`color(white)(xxxxxxxxxxxxxxx)=color(white)(xx)-6x^2-6bx-6c`

`color(white)(xxxxxxxxxxxxxxx)=x^3+color(blue)((b-6))x^2+color(red)((c-6b))x-6c`

we now compare coefficients and solve

(Coefficients of `x^2)`

`LHS=-10`

`RHS=b-6`

`color(blue)(b-6=-10=>b=-10+6)`

`b=-4`

Coefficients of x

`LHS=19`

`RHS=c-6b`

`color(red)(-6xx-4+c=19)`

`=>24+c=19`

`c=-5`

check constant term

`LHS=30`

`RHS=-6 xx -5=30`

`:.x^3-10x^2+19x+30=(x-6)(x^2-4x-5)`

now see if the quadratic factorises

factors `-5` that sum to `-4`

`-5" "`&`" "=+1`

we have therefore

`:.x^3-10x^2+19x+30=(x-6)(x-5)(x+1)`