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 find the other factor of the polynomial \( x^3 - 5x^2 - 2x + 24 \) given that \( x - 3 \) is a factor, we can use polynomial long division or synthetic division.

Since \( x - 3 \) is a factor, we can perform synthetic division with the root \( x = 3 \):

1. Write the coefficients of the polynomial: \( 1, -5, -2, 24 \).
2. Set up synthetic division:

\[
\begin{array}{r|rrrr}
3 & 1 & -5 & -2 & 24 \\
& & 3 & -6 & -24 \\
\hline
& 1 & -2 & -8 & 0 \\
\end{array}
\]

The result of synthetic division gives us the polynomial \( x^2 - 2x - 8 \).

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

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

Thus, the complete factorization of \( x^3 - 5x^2 - 2x + 24 \) is:

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

From the options provided, \( x + 2 \) is indeed a factor.

So the answer is:

**x + 2** (x plus 2)