Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x³−11x²+7x−5;f(4)

Verified step by step guidance

1. Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = 2x^{3} - 11x^{2} + 7x - 5\) and you want to find \(f(4)\).
2. Set up synthetic division by writing the coefficients of the polynomial in order: 2, -11, 7, -5. Use the value 4 as the divisor (since you are evaluating at \(x=4\)).
3. Begin synthetic division by bringing down the first coefficient (2) as is. Then multiply this number by 4 and write the result under the next coefficient.
4. Add the numbers in the second column, then repeat the multiply and add process for the remaining coefficients.
5. The final number you get after completing synthetic division is the remainder, which equals \(f(4)\) 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 it faster and less error-prone, especially for evaluating polynomials at specific values.

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 evaluating the polynomial at x = c gives the remainder directly, which is useful for quickly finding function values without full polynomial evaluation.

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a specific input. Using synthetic division or direct substitution, one can efficiently compute f(c) for a given c, which is essential for understanding function behavior and solving related algebra problems.