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? (1 point) The equation has all real solutions. The equation has two real solutions and one non-real solution. The equation has one real solution and two non-real solutions. The equation has all non-real solutions.

Answer 1

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

Step 1: Synthetic Division

We will perform synthetic division using \(x + 4\).

\[ \begin{array}{r|rrr} -4 & 1 & 1 & -16 & -16 \ & & -4 & 12 & 16 \ \hline & 1 & -3 & -4 & 0 \ \end{array} \]

The result of the synthetic division gives us the polynomial:

\[ x^2 - 3x - 4 \]

Step 2: Factor \(x^2 - 3x - 4\)

Next, we can factor \(x^2 - 3x - 4\):

\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]

Step 3: Combine Solutions

Therefore, the factored form of the original polynomial is:

\[ (x + 4)(x - 4)(x + 1) = 0 \]

Setting each factor to zero gives us the solutions:

1. \(x + 4 = 0 \Rightarrow x = -4\)
2. \(x - 4 = 0 \Rightarrow x = 4\)
3. \(x + 1 = 0 \Rightarrow x = -1\)

Step 4: Solution Set

The full solution set is:

\[ {-4, 4, -1} \]

Conclusion

All solutions are real numbers. Therefore, the correct statement that describes the solution set is:

The equation has all real solutions.