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)
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.
To factor the cubic polynomial f(x) = x^3 - 3x^2 - 10x + 24, let's first apply the Factor Theorem.
The Factor Theorem states that if a polynomial f(x) has a factor (x - c), then c is a zero of f(x). Since x = 2 is given as a zero of f(x), we can conclude that (x - 2) is a factor of f(x).
To factor f(x) into linear factors, we can divide f(x) by (x - 2) using the long division method or synthetic division. Let's use synthetic division:
2 | 1 -3 -10 24
-------------------
1 -1 -12 0
The result is 1x^2 - 1x - 12, with a remainder of 0.
Now, we need to factor 1x^2 - 1x - 12. This is a quadratic equation, which we can factor by looking for two numbers whose product is -12 and whose sum is -1. These numbers are -4 and 3.
Therefore, 1x^2 - 1x - 12 can be factored as (x - 4)(x + 3).
Putting it all together, we have:
f(x) = (x - 2)(x - 4)(x + 3)
So, the correct answer is option 3. f(x) = (x - 2)(x - 4)(x + 3).
To apply the Factor Theorem, we need to check if f(x) = x^3 - 3x^2 - 10x + 24 has x = 2 as a root.
We can do this by substituting x = 2 into f(x) and seeing if it equals to 0:
f(2) = (2)^3 - 3(2)^2 - 10(2) + 24
= 8 - 12 - 20 + 24
= 0
Since f(2) = 0, x = 2 is indeed a zero of f(x).
Now we can factor f(x) into linear factors using the zero x = 2 as a starting point.
The correct answer is:
3. f(x) = (x - 2)(x - 4)(x + 3)