Question

Determine the remaining linear factors of x4+7x3−x2−7x with known factors of x and (x+7) .(1 point) Responses (x+1) and (x−1) left parenthesis x plus 1 right parenthesis and left parenthesis x minus 1 right parenthesis (x−1) and (x−1) left parenthesis x minus 1 right parenthesis and left parenthesis x minus 1 right parenthesis (x+1) and (x+1) left parenthesis x plus 1 right parenthesis and left parenthesis x plus 1 right parenthesis (x2−1) and (x+1)

Answer 1

To find the remaining linear factors of the polynomial \( x^4 + 7x^3 - x^2 - 7x \) given the known factors \( x \) and \( (x + 7) \), we start by factoring the polynomial.

1. **Start with the given polynomial:**
\[
x^4 + 7x^3 - x^2 - 7x
\]

2. **Factor out the known factors \( x \) and \( (x + 7) \):**
Since one factor is \( x \), we can factor \( x \) out:
\[
x(x^3 + 7x^2 - x - 7)
\]

3. **Next, we will factor \( x^3 + 7x^2 - x - 7 \)**. Since we know another factor is \( (x + 7) \), we can perform polynomial long division or synthetic division to divide \( x^3 + 7x^2 - x - 7 \) by \( (x + 7) \).

- **Using synthetic division:**

\[
\begin{array}{r|rrrr}
-7 & 1 & 7 & -1 & -7 \\
& & -7 & 0 & 7 \\
\hline
& 1 & 0 & -1 & 0 \\
\end{array}
\]
The result of this division is:
\[
x^2 - 1
\]

4. **Thus, we can write the polynomial as:**
\[
x(x + 7)(x^2 - 1)
\]

5. **Next, we factor \( x^2 - 1 \) further:**
\[
x^2 - 1 = (x - 1)(x + 1)
\]

6. **Putting all the factors together:**
\[
x(x + 7)(x - 1)(x + 1)
\]

Now, the linear factors of the polynomial \( x^4 + 7x^3 - x^2 - 7x \) are \( x \), \( (x + 7) \), \( (x - 1) \), and \( (x + 1) \).

7. **Therefore, the remaining linear factors are \( (x + 1) \) and \( (x - 1) \)**.

The correct response is:
**(x + 1) and (x - 1)**.