For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = - x³ + 8x² + 63; k=4

Verified step by step guidance

1. Recall the Remainder Theorem, which states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). So, to find ƒ(4), we simply substitute x = 4 into the polynomial.
2. Write down the polynomial function: \(ƒ(x) = -x^{3} + 8x^{2} + 63\).
3. Substitute \(x = 4\) into the polynomial: \(ƒ(4) = -(4)^{3} + 8(4)^{2} + 63\).
4. Calculate each term separately: compute \(-(4)^{3}\), then \$8(4)^{2}$, and finally add 63.
5. Add the results of these calculations to find the value of \(ƒ(4)\), which is the remainder when dividing by \((x - 4)\).

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of polynomial functions helps in evaluating them at specific values and applying related theorems.

Remainder Theorem

The Remainder Theorem states that when a polynomial ƒ(x) is divided by (x - k), the remainder is equal to ƒ(k). This theorem allows us to find the value of the polynomial at k without performing full polynomial division.

Evaluating Polynomials

Evaluating a polynomial at a given value involves substituting the variable with that value and simplifying. This process is essential for applying the Remainder Theorem and finding ƒ(k) efficiently.