For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 2x⁵ - 10x³ - 19x² - 50; k=3

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(3), we simply evaluate the polynomial at x = 3.
2. Write down the polynomial function: \(ƒ(x) = 2x^{5} - 10x^{3} - 19x^{2} - 50\).
3. Substitute \(x = 3\) into the polynomial: \(ƒ(3) = 2(3)^{5} - 10(3)^{3} - 19(3)^{2} - 50\).
4. Calculate each term separately: compute \$2(3)^{5}\(, \)-10(3)^{3}\(, \)-19(3)^{2}\(, and \)-50$.
5. Add all the calculated terms together to find the value of \(ƒ(3)\), which is the remainder when dividing by \((x - 3)\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. 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 theorem allows for quick evaluation of the polynomial at x = k without performing full polynomial division.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value into the polynomial and simplifying. This process is essential for applying the Remainder Theorem and finding the remainder or value of the function at that point.