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)

Verified step by step guidance

1. Identify the divisor for synthetic division, which is x - (-3) or x + 3.
2. Set up the synthetic division by writing the coefficients of the polynomial f(x) = 3x^3 - 7x^2 - 2x + 5, which are 3, -7, -2, and 5.
3. Write -3 (the value of x for f(-3)) to the left of the coefficients.
4. Bring down the leading coefficient (3) to the bottom row.
5. Multiply -3 by the number just written on the bottom row (3), and write the result under the next coefficient (-7). Add this result to -7 and write the sum below. Repeat this process for the remaining coefficients.

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 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.

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 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.

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.