Question

Which of the following is a factor of x3−5x2−2x+24
, given that x−3
is one of the linear factors?(1 point)
Responses

x2−8x+22
x squared minus 8 x plus 22

x−2
x minus 2

x+2
x plus 2

x+4

Answer 1

To determine which of the options is a factor of \( x^3 - 5x^2 - 2x + 24 \), given that \( x - 3 \) is a factor, we can use polynomial long division to divide \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \).

1. **Long Division**:
- Divide the leading term: \( x^3 \div x = x^2 \).
- Multiply \( x^2 \) by \( x - 3 \): \( x^3 - 3x^2 \).
- Subtract: \((x^3 - 5x^2) - (x^3 - 3x^2) = -2x^2\).
- Bring down the next term: \(-2x^2 - 2x = -2x^2 - 2x\).
- Divide \(-2x^2\) by \(x\): \(-2x\).
- Multiply: \(-2x(x - 3) = -2x^2 + 6x\).
- Subtract: \((-2x^2 - 2x) - (-2x^2 + 6x) = -8x\).
- Bring down the next term: \(-8x + 24\).
- Divide: \(-8x \div x = -8\).
- Multiply: \(-8(x - 3) = -8x + 24\).
- Subtract: \((-8x + 24) - (-8x + 24) = 0\).

Thus, we find that:

\[
x^3 - 5x^2 - 2x + 24 = (x - 3)(x^2 - 2x - 8)
\]

Now, we can factor \( x^2 - 2x - 8 \).

2. **Factoring**:
To factor \( x^2 - 2x - 8 \):
- We need two numbers that multiply to \(-8\) (the constant term) and add to \(-2\) (the coefficient of \( x \)). These numbers are \(-4\) and \(2\).

So,

\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

3. **Finding the Factors**:
Thus, the complete factorization of the polynomial is:

\[
x^3 - 5x^2 - 2x + 24 = (x - 3)(x - 4)(x + 2)
\]

Since \( x + 2 \) is one of the factors, the answer is:

**x + 2**.