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

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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 substitute k into the polynomial.

Identify the polynomial and the value of k: Here, ƒ(x) = x^2 - 5x + 1 and k = 2 + i, where i is the imaginary unit.

Substitute k = 2 + i into the polynomial: Replace every x in ƒ(x) with (2 + i), so write ƒ(2 + i) = (2 + i)^2 - 5(2 + i) + 1.

Expand the squared term: Calculate (2 + i)^2 by using the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 2 and b = i.

Simplify the expression by performing the multiplication and combining like terms, remembering that i^2 = -1.

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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, such as complex numbers.

Remainder Theorem

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

Evaluating Polynomials at Complex Numbers

Evaluating a polynomial at a complex number involves substituting the complex value into the polynomial and simplifying using algebraic rules for complex numbers, including combining like terms and applying i² = -1. This is essential when k is a complex number like 2 + i.

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