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

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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.

Write down the polynomial function: $ƒ(x) = -x^{3} + 8x^{2} + 63$.

Substitute $x = 4$ into the polynomial: $ƒ(4) = -(4)^{3} + 8(4)^{2} + 63$.

Calculate each term separately: compute $-(4)^{3}$, then \$8(4)^{2}$, and finally add 63.

Add the results of these calculations to find the value of $ƒ(4)$, which is the remainder when dividing by $(x - 4)$.

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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, 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.

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