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
f(x) = (x - 2)(x - 3)(x - 4)
f(x) = (x + 2)(x + 3)(x - 4)
f(x) = (x - 2)(x^2 - x - 12)
f(x) = (x - 2)(x + 3)(x - 4)
The correct answer is f(x) = (x - 2)(x + 3)(x - 4).
To apply the factor theorem and factor f(x) into linear factors, we need to find the factors of f(x) based on the given zero x = 2.
First, we use the factor theorem, which states that if x - c is a factor of f(x), then f(c) = 0. Since x = 2 is a zero of f(x), we can substitute x = 2 into f(x) and check if it equals to 0.
f(2) = (2)^3 - 3(2)^2 - 10(2) + 24
f(2) = 8 - 12 - 20 + 24
f(2) = 0
Since f(2) = 0, we can conclude that x - 2 is a factor of f(x).
To determine the other factors, we divide f(x) by (x - 2) using long division or synthetic division.
(x^3 - 3x^2 - 10x + 24) / (x - 2)
Performing long division or synthetic division, we find that the quotient is (x^2 + x - 12).
Therefore, f(x) can be factored as (x - 2)(x^2 + x - 12).
To further factorize the quadratic term (x^2 + x - 12), we can apply the factoring method to find two linear factors.
By factoring the quadratic term, we have:
(x - 2)(x + 4)(x - 3)
Therefore, f(x) can be factored into linear factors as:
f(x) = (x - 2)(x + 4)(x - 3)
To determine the factors of a polynomial, we can use the Factor Theorem. According to the Factor Theorem, if x = c is a zero of a polynomial f(x), then (x - c) is a factor of f(x).
Given that x = 2 is a zero of f(x) = x^3 - 3x^2 - 10x + 24, we can apply the Factor Theorem.
First, 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 quotient is x^2 - x - 12, and there is no remainder.
Now, we can factor x^2 - x - 12 into linear factors.
To do this, we need to find two numbers whose product is -12 and whose sum is -1 (the coefficient of the x term).
The numbers that satisfy these conditions are -4 and 3, which means the factors of x^2 - x - 12 are (x - 4)(x + 3).
Therefore, factoring f(x) into linear factors, we get:
f(x) = (x - 2)(x - 4)(x + 3)