In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x^3−11x^2+7x−5;f(4)

Verified step by step guidance

1. Identify the divisor for synthetic division, which is x - 4, since we are evaluating f(4).
2. Set up the synthetic division by writing the coefficients of the polynomial f(x) = 2x^3 - 11x^2 + 7x - 5, which are 2, -11, 7, and -5.
3. Write the value 4 (from x - 4) to the left of the coefficients and draw a horizontal line below them.
4. Bring down the leading coefficient (2) to the bottom row.
5. Multiply the value 4 by the number just written below the line (2), and write the result under the next coefficient (-11). Continue this process of multiplying and adding down the column until you reach the last coefficient.

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 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 fully performing the division, making it a powerful tool for quickly finding function values and understanding polynomial behavior.

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 - 11x^2 + 7x - 5 is a cubic polynomial, which can have up to three real roots and exhibits a continuous curve. Understanding the structure of polynomial functions is essential for applying synthetic division and the Remainder Theorem effectively.