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EXAMPLE 3

Write the coefficients of f ( x ) in order of descending exponents. Write the value at which f ( x ) is being evaluated to the left. STEP 1. EXAMPLE 3. Evaluate by synthetic substitution.

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EXAMPLE 3

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  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. STEP 1 EXAMPLE 3 Evaluate by synthetic substitution Use synthetic substitution to evaluate f (x) from Example2 when x = 3.f (x) = 2x4 – 5x3 –4x + 8 SOLUTION

  2. STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add. STEP 3 Multiply 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. EXAMPLE 3 Evaluate by synthetic substitution

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

  4. ANSWER The correct answer is D. 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.

  5. for Examples 3 and 4 GUIDED PRACTICE Use synthetic substitution to evaluate the polynomial function for the given value of x. 6. f (x) = 5x3 + 3x2–x + 7; x = 2 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left. STEP 1

  6. STEP 3 Multiply 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. for Examples 3 and 4 GUIDED PRACTICE STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add.

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

  8. – 1 – 2 – 1 0 4 – 5 for Examples 3 and 4 GUIDED PRACTICE 7. g (x) = – 2x4–x3 + 4x – 5; x = – 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. STEP 1

  9. – 2 1 – 1 – 2 – 1 0 4 – 5 STEP 3 Multiply 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. for Examples 3 and 4 GUIDED PRACTICE STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add.

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

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

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