Question

Let `f(x)=x^3 +mx^2 +5x -n` and `g(x)=x^3-x^2-(m+2)x +n`, where `m` and `n` are cosntants. It is given that `x+3` is a common factor of `f(x)` and `g(x)`, find the value of `m` and `n` ?

Answer 1

`m = 6" "` and `" "n = 12`

Explanation:

`(x+3)` is a factor if and only if `x=-3` is a zero.

So we have:

`0 = f(-3) = (color(blue)(-3))^3+m(color(blue)(-3))^2+5(color(blue)(-3))-n`

`color(white)(0 = f(-3)) = -27+9m-15-n`

`color(white)(0 = f(-3)) = -42+9m-n`

`0 = g(-3) = (color(blue)(-3))^3-(color(blue)(-3))^2-(m+2)(color(blue)(-3))+n`

`color(white)(0 = g(-3)) = -27-9+3m+6+n`

`color(white)(0 = g(-3)) = -30+3m+n`

Adding these two equations, we get:

`0 = -72+12m`

Hence:

`m = 6`

Then:

`n = 9m-42 = 54-42 = 12`