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

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 to find the quotient and remainder. This technique is particularly useful for quickly evaluating polynomials at specific values, as it reduces the computational complexity compared to long division.

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 allows us to evaluate the polynomial at a specific point without performing the entire division process. In this case, it provides a direct way to find f(-3) by using synthetic division to determine the remainder.

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.