In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x^4−5x^3−x^2+3x+2;f(− 1/2)

Verified step by step guidance

1. Identify the polynomial function \( f(x) = 2x^4 - 5x^3 - x^2 + 3x + 2 \) and the value \( x = -\frac{1}{2} \) for which we need to find \( f(-\frac{1}{2}) \).
2. Set up synthetic division using \( -\frac{1}{2} \) as the divisor. Write down the coefficients of the polynomial: \( 2, -5, -1, 3, 2 \).
3. Bring down the leading coefficient (2) to the bottom row.
4. Multiply the divisor \( -\frac{1}{2} \) by the number just written on the bottom row (2) and write the result under the next coefficient (-5).
5. Add the numbers in the second column: \(-5 + (-1) = -6\). Repeat the multiplication and addition process for the remaining coefficients. The final number in the bottom row is the remainder, which is \( f(-\frac{1}{2}) \).

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions, which allows for quicker calculations compared to long division. This technique is particularly useful for evaluating polynomials at specific values, as it can directly yield the remainder, which corresponds to the function value at that point.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x - c), the remainder of this division is equal to f(c). This theorem provides a quick way to evaluate the polynomial at a specific point without fully performing the division. In the context of the given problem, using the Remainder Theorem allows us to find f(-1/2) efficiently by applying synthetic division.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable x is f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n are constants and n is a non-negative integer. Understanding the structure of polynomial functions is essential for applying synthetic division and the Remainder Theorem effectively, as it helps in recognizing the degree and behavior of the function.