Question

What are the remaining linear factors of x^4−13x^3−x^2+13x with known factors (x+1) and (x−1) ?(1 point)

Answer 1

To find the remaining linear factors of the polynomial \( x^4 - 13x^3 - x^2 + 13x \) given that \( (x + 1) \) and \( (x - 1) \) are factors, we can start by performing polynomial division.

1. First, we can factor the polynomial:
\[
x^4 - 13x^3 - x^2 + 13x = x(x^3 - 13x^2 - x + 13)
\]

2. Now, we need to perform synthetic division on \( x^3 - 13x^2 - x + 13 \) using the known roots \( x = -1 \) and \( x = 1 \).

**Synthetic Division by \( x + 1 \):**

Using \( x = -1 \):
\[
\begin{array}{r|rrrr}
-1 & 1 & -13 & -1 & 13 \\
& & -1 & 14 & -13 \\
\hline
& 1 & -14 & 13 & 0 \\
\end{array}
\]
This gives us a quotient of \( x^2 - 14x + 13 \).

**Then we divide \( x^2 - 14x + 13 \) by \( x - 1 \):**

Using \( x = 1 \):
\[
\begin{array}{r|rr}
1 & 1 & -14 & 13 \\
& & 1 & -13 \\
\hline
& 1 & -13 & 0 \\
\end{array}
\]
This gives us a quotient of \( x - 13 \).

The result of the full factorization is:
\[
x^4 - 13x^3 - x^2 + 13x = x(x + 1)(x - 1)(x - 13)
\]

So, the remaining linear factor of the polynomial is \( (x - 13) \).

Thus, the remaining linear factors of the polynomial are \( (x + 1) \), \( (x - 1) \), and \( (x - 13) \).