In Exercises 33–40, 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 $f\left(4\right)$.
2. Set up synthetic division by writing the coefficients of the polynomial in descending order of powers: 2, -11, 7, -5.
3. Use the value 4 (from $f\left(4\right)$) as the divisor in synthetic division. Write 4 to the left and draw a horizontal line to separate the work area.
4. Bring down the first coefficient (2) as it is. Multiply this number by 4 and write the result under the next coefficient. Add the column and write the sum below the line. Repeat this multiply-and-add process for all coefficients.
5. The final number you get after completing synthetic division is the remainder, which equals $f\left(4\right)$ 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 division process by using only the coefficients of the polynomial, making calculations faster and less error-prone compared to long division.

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 or checking factors.

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a specific input x = c. Using synthetic division or direct substitution, you can compute f(c), which helps in understanding the behavior of the polynomial and solving related problems.