Question

One of the solutions to the equation x^3+x^2−16x−16=0 is −4 . Test the solution to find the full solution set. Which of the following statements correctly describes the solution set?

Answer 1

To find the full solution set of the equation \( x^3 + x^2 - 16x - 16 = 0 \) given that one of the solutions is \( x = -4 \), we can use synthetic division to factor the polynomial.

Since \( x = -4 \) is a root, we can use synthetic division with -4:

Set up synthetic division:
```
-4 | 1 1 -16 -16
| -4 12 16
----------------------
1 -3 -4 0
```

The quotient is:
\[
x^2 - 3x - 4
\]

Now, we need to factor the quadratic \( x^2 - 3x - 4 \). We can do this by looking for two numbers that multiply to \(-4\) and add to \(-3\). The numbers \(-4\) and \(1\) satisfy these conditions.

Thus, we can factor it as:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]

So, the complete factorization of the original cubic polynomial is:
\[
x^3 + x^2 - 16x - 16 = (x + 4)(x - 4)(x + 1)
\]

Now, we can set each factor equal to zero to find all the solutions:
1. \( x + 4 = 0 \rightarrow x = -4 \)
2. \( x - 4 = 0 \rightarrow x = 4 \)
3. \( x + 1 = 0 \rightarrow x = -1 \)

Thus, the full solution set is:
\[
\{ -4, 4, -1 \}
\]

**Conclusion:**
The statement that correctly describes the solution set is that the solutions of the equation \( x^3 + x^2 - 16x - 16 = 0 \) are \( -4, 4, \) and \( -1 \).