Question

Factor the polynomial f(x). Then solve the equation f(x)=0.
f(x)=x^3-3x^2-10x+24

Answer 1

By Rational Root Theorem, the likely zeroes are, 1,2,3,4,6,8,12,24. After a bit of trial and error, we see that x = 2, is a zero, therefore x - 2 = 0 and x - 2 is a factor of f(x).

Answer 2

Next, Take f(x) and split it in accordance with the known factor x - 2, then factor the remaining quadratic as follows:

x^3 - 3x^2 - 10x + 24 = x^3 - 2x^2 - x^2 + 2x - 12x + 24
= x^2(x - 2) - x(x - 2) - 12(x - 2)
= (x - 2)(x^2 - x - 12) <---remaining quadratic
= (x - 2)(x - 4)(x + 3)

Answer 3

So f(x) = (x - 2)(x - 4)(x + 3)

Now set f(x) = 0

0 = (x - 2)(x - 4)(x + 3)

By Zero Product Property:

x - 2 = 0
x - 4 = 0
x + 3 = 0

You should be able to find x from there.

Answer 4

Any questions?

Answer 5

Thank you

Answer 6

You're welcome