To determine the factors of the polynomial [tex]\( p(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \)[/tex] using synthetic division, we need to test potential rational roots. These potential roots are derived from the factors of the constant term ([tex]\( \pm 24 \)[/tex]) and the leading coefficient ([tex]\( 1 \)[/tex]), which are: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \)[/tex].
We evaluate each candidate factor, and if the remainder of the synthetic division is zero, then the candidate is a root of the polynomial. Let's walk through the synthetic division process for each valid root.
### Step-by-Step Synthetic Division Process
#### Testing [tex]\( x = -4 \)[/tex]:
1. Write down the coefficients of [tex]\( p(x) \)[/tex]: [tex]\( 1, 4, -7, -22, 24 \)[/tex].
2. Set up the synthetic division:
[tex]\[
\begin{array}{r|rrrrr}
-4 & 1 & 4 & -7 & -22 & 24 \\
& & -4 & 0 & 28 & -24 \\
\hline
& 1 & 0 & -7 & 6 & 0 \\
\end{array}
\][/tex]
Since the remainder is [tex]\( 0 \)[/tex], [tex]\( x = -4 \)[/tex] is a root.
#### Testing [tex]\( x = -3 \)[/tex]:
1. Using the same coefficients [tex]\( 1, 4, -7, -22, 24 \)[/tex]:
2. Synthetic division setup:
[tex]\[
\begin{array}{r|rrrrr}
-3 & 1 & 4 & -7 & -22 & 24 \\
& & -3 & -3 & 30 & -24 \\
\hline
& 1 & 1 & -10 & 8 & 0 \\
\end{array}
\][/tex]
The remainder [tex]\( 0 \)[/tex], so [tex]\( x = -3 \)[/tex] is a root.
#### Testing [tex]\( x = 1 \)[/tex]:
1. Using the coefficients [tex]\( 1, 4, -7, -22, 24 \)[/tex]:
2. Synthetic division setup:
[tex]\[
\begin{array}{r|rrrrr}
1 & 1 & 4 & -7 & -22 & 24 \\
& & 1 & 5 & -2 & -24 \\
\hline
& 1 & 5 & -2 & -24 & 0 \\
\end{array}
\][/tex]
The remainder [tex]\( 0 \)[/tex], so [tex]\( x = 1 \)[/tex] is a root.
#### Testing [tex]\( x = 2 \)[/tex]:
1. Using the coefficients [tex]\( 1, 4, -7, -22, 24 \)[/tex]:
2. Synthetic division setup:
[tex]\[
\begin{array}{r|rrrrr}
2 & 1 & 4 & -7 & -22 & 24 \\
& & 2 & 12 & 10 & -24 \\
\hline
& 1 & 6 & 5 & -12 & 0 \\
\end{array}
\][/tex]
The remainder [tex]\( 0 \)[/tex], so [tex]\( x = 2 \)[/tex] is a root.
### Conclusion
We have identified through synthetic division that the factors of the polynomial [tex]\( p(x) \)[/tex] are [tex]\( x = -4, -3, 1, 2 \)[/tex].
Thus, the correct roots using synthetic division are:
[tex]\[
\boxed{x = -4, x = -3, x = 1, x = 2}
\][/tex]
