For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 6x⁴ + x³ - 8x² + 5x+6; k=1/2

Verified step by step guidance

1. Recall the Remainder Theorem: For a polynomial ƒ(x), the remainder when divided by (x - k) is equal to ƒ(k). So, to find ƒ(k), we simply evaluate the polynomial at x = k.
2. Write down the polynomial function and substitute x with k = \(\frac{1}{2}\): \[ƒ\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^3 - 8\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) + 6\]
3. Calculate each term separately: - Compute \(6\left(\frac{1}{2}\right)^4\) - Compute \(\left(\frac{1}{2}\right)^3\) - Compute \(-8\left(\frac{1}{2}\right)^2\) - Compute \(5\left(\frac{1}{2}\right)\) - The constant term is 6
4. Add all the computed values together to find the value of ƒ\(\left\)(\(\frac{1}{2}\)\(\right\)). This sum represents the remainder when ƒ(x) is divided by \(x - \frac{1}{2}\).
5. Thus, the value of ƒ(k) is the remainder according to the Remainder Theorem, completing the problem.

Key Concepts

Polynomial Functions

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

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 allows us to find the value of the polynomial at k by simply evaluating ƒ(k), without performing full polynomial division.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value for the variable and simplifying the expression. This process is essential for applying the Remainder Theorem and finding the remainder or function value efficiently.