Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).

Verified step by step guidance

1. Identify the polynomial function given: \( f(x) = 2x^3 - 7x^2 + 9x - 3 \).
2. According to the Remainder Theorem, the remainder of the division of \( f(x) \) by \( x - c \) is \( f(c) \).
3. In this problem, we need to find \( f(-13) \), so \( c = -13 \).
4. Substitute \( x = -13 \) into the polynomial: \( f(-13) = 2(-13)^3 - 7(-13)^2 + 9(-13) - 3 \).
5. Calculate each term separately and then sum them to find the remainder, which is \( f(-13) \).

Key Concepts

Remainder Theorem

The Remainder Theorem states that for a polynomial function f(x), the remainder of the division of f(x) by (x - c) is equal to f(c). This theorem allows us to evaluate the polynomial at a specific point without performing long division, simplifying the process of finding function values.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = 2x^3 - 7x^2 + 9x - 3 is a cubic polynomial, which means it has a degree of 3 and can have up to three real roots. Understanding the structure of polynomial functions is essential for applying the Remainder Theorem.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this context, evaluating f(-13) means replacing x in the polynomial with -13 and calculating the resulting value. This process is fundamental in applying the Remainder Theorem to find the remainder when the polynomial is evaluated at a given point.