To determine the full solution set for the polynomial equation \( x^3 + x^2 - 16x - 16 = 0 \) given that one of the solutions is \( x = -4 \), we can use polynomial long division or synthetic division to factor the polynomial.
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Using Synthetic Division with \( x = -4 \):
We set up synthetic division as follows:
-4 | 1 1 -16 -16 | -4 12 16 --------------------- 1 -3 -4 0The result is \( x^2 - 3x - 4 \) with a remainder of 0. This means the polynomial factors as:
\[ (x + 4)(x^2 - 3x - 4) \]
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Factoring the Quadratic:
Now, we need to factor \( x^2 - 3x - 4 \). We look for two numbers that multiply to \(-4\) and add to \(-3\), which are \(-4\) and \(1\). Thus:
\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]
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Finding the Complete Factorization:
Combining everything, the original polynomial factors as:
\[ (x + 4)(x - 4)(x + 1) = 0 \]
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Finding the Solutions:
Setting each factor to zero gives us the solutions:
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
Thus, the complete solution set is \( x = -4, 4, -1 \).
- Describing the Solution Set:
All three solutions (-4, 4, and -1) are real numbers. Therefore, the correct option from the provided choices is:
The equation has all real solutions.
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