In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x^4+5x^3+5x^2−5x−6;f(3)

Verified step by step guidance

1. Identify the polynomial function \( f(x) = x^4 + 5x^3 + 5x^2 - 5x - 6 \) and the value \( x = 3 \) for which we need to find \( f(3) \).
2. Set up synthetic division by writing the coefficients of the polynomial: \( 1, 5, 5, -5, -6 \).
3. Write the value \( 3 \) to the left of the coefficients, as this is the value for which we are evaluating the function.
4. Bring down the leading coefficient (1) to the bottom row.
5. Multiply the value brought down (1) by \( 3 \) and write the result under the next coefficient (5). Add this result to the coefficient and continue this process across all 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 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 factor (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) = x^4 + 5x^3 + 5x^2 - 5x - 6 is a fourth-degree polynomial. Understanding the structure of polynomial functions, including their degree, leading coefficient, and behavior at various points, is essential for applying synthetic division and the Remainder Theorem effectively.