To find $f(-5)$ and $f(2)$ for the polynomial function $f(x) = x^5 + 8x^3 + 2x - 15$ using synthetic substitution, follow these steps:

Step 1: Set up the coefficients

The polynomial is: $f(x) = x^5 + 8x^3 + 2x - 15$ The coefficients of the polynomial are: $[1, 0, 8, 0, 2, -15]$

Step 2: Synthetic substitution for $f(-5)$

1. Write $-5$ as the divisor (the value for substitution).
2. Perform synthetic division using the coefficients $[1, 0, 8, 0, 2, -15]$.

-5 & 1 & 0 & 8 & 0 & 2 & -15 \\ & & -5 & 25 & -165 & 825 & -4110 \\ \hline & 1 & -5 & 33 & -165 & 827 & -4125 \\ \end{array}$$ The result is $$f(-5) = -4125$$. --- ### Step 3: Synthetic substitution for $$f(2)$$ 1. Write $$2$$ as the divisor (the value for substitution). 2. Perform synthetic division using the coefficients $$[1, 0, 8, 0, 2, -15]$$. $$\begin{array}{r|rrrrrr} 2 & 1 & 0 & 8 & 0 & 2 & -15 \\ & & 2 & 4 & 24 & 48 & 100 \\ \hline & 1 & 2 & 12 & 24 & 50 & 85 \\ \end{array}$$ The result is $$f(2) = 85$$. --- ### Final Results: $$f(-5) = -4125$$ $$f(2) = 85$$ Would you like a detailed explanation of the synthetic substitution steps? ### Related Questions: 1. How does synthetic substitution differ from direct evaluation of polynomials? 2. Can synthetic substitution be used for non-integer inputs like $$f(1.5)$$? 3. What are the advantages of synthetic substitution over traditional substitution? 4. How can the remainder theorem help verify these results? 5. Can this process be extended to find derivatives at a specific point? **Tip:** Synthetic substitution is particularly useful for quick evaluation of polynomials when testing multiple roots or checking for divisors.