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

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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.

Write down the polynomial function and substitute x with k = \(\frac{1}{2}\): \[ƒ\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^3 - 8\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) + 6\]

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

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 \(x - \frac{1}{2}\).

Thus, the value of ƒ(k) is the remainder according to the Remainder Theorem, completing the problem.

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주요 개념

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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.

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