1. EXAMPLE 3 Evaluate by synthetic substitution Use synthetic substitution to evaluate f (x) from Example 2 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 SOLUTION STEP 1 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

2. EXAMPLE 3 Evaluate by synthetic substitution STEP 2Bring down the leading coefficient. Multiply the leading coefficient by the x -value. Write the product under the second coefficient. Add. STEP 3Multiply the previous sum by the x- value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x -value.

3. EXAMPLE 3 Evaluate by synthetic substitution Synthetic substitution gives f(3) = 23, which matches the result in Example 2. ANSWER

4. EXAMPLE 4 Standardized Test Practice From the graph, f (x) → – ∞ as x → – ∞ and f (x) → – ∞ as x → + ∞. So, the degree is even and the leading coefficient is negative. The correct answer is D.ANSWER

5. GUIDED PRACTICE for Examples 3 and 4 Use synthetic substitution to evaluate the polynomial function for the given value of x. 6. f (x) = 5x 3 + 3x 2 – x + 7; x = 2 STEP 1 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

6. GUIDED PRACTICE for Examples 3 and 4 STEP 2Bring down the leading coefficient. Multiply the leading coefficient by the x -value. Write the product under the second coefficient. Add. STEP 3Multiply the previous sum by the x- value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x -value.

7. GUIDED PRACTICE for Examples 3 and 4 Synthetic substitution gives f(2) = 57 ANSWER

8. GUIDED PRACTICE for Examples 3 and 4 7. g (x) = – 2x 4 – x 3 + 4x – 5; x = – 1 STEP 1 Write the coefficients of g(x) in order of descending exponents. Write the value at which g (x) is being evaluated to the left. – 1 – 2 – 1 0 4 – 5

9. GUIDED PRACTICE for Examples 3 and 4 STEP 2Bring down the leading coefficient. Multiply the leading coefficient by the x -value. Write the product under the second coefficient. Add. – 1 – 2 – 1 0 4 – 5 – 2 1 STEP 3Multiply the previous sum by the x- value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x -value.

10. GUIDED PRACTICE for Examples 3 and 4 2 –1 1 –10 – 1 – 2 – 1 0 4 – 5 – 2 1 –1 5 – 10 Synthetic substitution gives f(– 1) = – 10 ANSWER

11. GUIDED PRACTICE for Examples 3 and 4 8. Describe the degree and leading coefficient of the polynomial function whose graph is shown. ANSWER degree: odd, leading coefficient: negative