In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x⁴−5x³−x²+3x+2;f(− 1/2)

Verified step by step guidance

1. Identify the divisor for synthetic division based on the value at which the function is evaluated. Since we want to find \( f\left(-\frac{1}{2}\right) \), the divisor is \( x - \left(-\frac{1}{2}\right) = x + \frac{1}{2} \).
2. Set up synthetic division using the coefficients of the polynomial \( f(x) = 2x^4 - 5x^3 - x^2 + 3x + 2 \). The coefficients are \( [2, -5, -1, 3, 2] \).
3. Write the value \( -\frac{1}{2} \) to the left of the synthetic division setup and bring down the first coefficient (2) as is.
4. Multiply \( -\frac{1}{2} \) by the number just brought down (2), write the result under the next coefficient, then add the column. Repeat this multiply-and-add process for each coefficient.
5. The final number obtained after completing synthetic division is the remainder, which equals \( f\left(-\frac{1}{2}\right) \) by the Remainder Theorem.

Key Concepts

Synthetic Division

Synthetic division is a streamlined 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 technique is especially useful 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 that synthetic division can be used to find the value of the polynomial at x = c by identifying the remainder after division, providing a quick way to evaluate polynomials.

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a given input. Using synthetic division and the Remainder Theorem, one can efficiently compute f(c) without substituting and calculating each term individually, which is especially helpful for higher-degree polynomials.