In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x³−7x²−2x+5;f(−3)

Verified step by step guidance

1. Identify the divisor for synthetic division based on the function value you want to find. Since you want to find f(−3), the divisor is x - (−3) or x + 3.
2. Set up synthetic division by writing the coefficients of the polynomial f(x) = 3x³ − 7x² − 2x + 5. These coefficients are 3, -7, -2, and 5.
3. Write the value −3 (from x + 3) to the left side of the synthetic division setup and bring down the first coefficient (3) as is.
4. Multiply −3 by the number just brought down (3), write the result under the next coefficient (-7), then add the column. Repeat this multiply and add process for each coefficient.
5. The final number you get after completing the synthetic division process is the remainder, which equals f(−3) by the Remainder Theorem.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone. This method is especially useful for evaluating polynomials and finding remainders.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means you can find the value of the polynomial at x = c by performing synthetic division and looking at the remainder, providing a quick way to evaluate polynomials without direct substitution.

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a specific input value. Instead of substituting directly, synthetic division combined with the Remainder Theorem offers an efficient alternative to compute f(c), especially for higher-degree polynomials, by reducing computational complexity.