Question

Given that x=2 is a zero of f(x)=x^3-3x^2-10x+24, apply the Factor Theorem and factor f(x) into linear factors. (2 points).

1. f(x)=(x-2)(x-3)(x-4)
2. f(x)=(x-2)(x^2-x-12)
3. f(x)=(x-2)(x-4)(x+3)
4. f(x)=(x+2)(x-4)(x+3)

Answer 1

To apply the Factor Theorem, we need to check if f(x) is equal to the product of (x - 2) and another factor.

Let's evaluate f(2):
f(2) = 2^3 - 3(2)^2 - 10(2) + 24
= 8 - 12 - 20 + 24
= 0

Since f(2) = 0, we can conclude that (x - 2) is a factor of f(x).

Now we can use long division or synthetic division to divide f(x) by (x - 2):
x^2 - x - 12
________________________
x - 2 | x^3 - 3x^2 - 10x + 24
- (x^3 - 2x^2)
________________________
-x^2 - 10x
+ (x^2 - 2x)
________________________
-8x + 24
+ (-8x + 16)
________________________
8

The remainder is 8, not 0. This means that (x - 2) is not a factor of f(x), and we cannot factor f(x) into linear factors.

Therefore, none of the given answer choices are correct.