Question

Question Given that x=2 is a zero of f(x)=x3−3x2−10x+24 , apply the Factor Theorem and factor f(x) into linear factors.(1 point) Responses f(x)=(x−2)(x2−x−12) f of x is equal to open paren x minus 2 close paren times open paren x squared minus x minus 12 close paren f(x)=(x−2)(x+3)(x−4) f of x is equal to open paren x minus 2 close paren times open paren x plus 3 close paren times open paren x minus 4 close paren f(x)=(x−2)(x−3)(x−4) f of x is equal to open paren x minus 2 close paren times open paren x minus 3 close paren times open paren x minus 4 close paren f(x)=(x+2)(x+3)(x−4)

Answer 1

To factor the polynomial \( f(x) = x^3 - 3x^2 - 10x + 24 \) using the Factor Theorem, we first confirm that \( x = 2 \) is indeed a zero. Then we can divide \( f(x) \) by \( (x - 2) \) to find the other factors.

### Step 1: Synthetic Division

Using synthetic division with \( x = 2 \):

```
2 | 1 -3 -10 24
| 2 -2 -24
---------------------------
1 -1 -12 0
```

The result is \( x^2 - x - 12 \), and since there's a zero remainder, we have:

\[
f(x) = (x - 2)(x^2 - x - 12)
\]

### Step 2: Factor the Quadratic

Next, we need to factor \( x^2 - x - 12 \). We look for two numbers that multiply to \(-12\) and add to \(-1\). The numbers \(-4\) and \(3\) fit this requirement:

\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

### Step 3: Combine All Factors

Now we can write \( f(x) \) as:

\[
f(x) = (x - 2)(x - 4)(x + 3)
\]

### Final Answer

Thus, the factored form of the polynomial \( f(x) \) into linear factors is:

\[
f(x) = (x - 2)(x + 3)(x - 4)
\]

Among the provided options, the correct response is:

**f(x) = (x - 2)(x + 3)(x - 4)**.