Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = x^2 - 2x + 2; k = 1-i

Verified step by step guidance

1. Identify the polynomial function \( f(x) = x^2 - 2x + 2 \) and the complex number \( k = 1-i \).
2. Set up synthetic division using \( k = 1-i \). Write the coefficients of the polynomial: \( 1, -2, 2 \).
3. Perform synthetic division: Bring down the leading coefficient (1) as the first entry in the bottom row.
4. Multiply \( 1 \) by \( k = 1-i \) and add the result to the next coefficient (-2).
5. Continue the process: Multiply the new entry by \( k = 1-i \) and add to the next coefficient (2).

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder. This technique is particularly useful for evaluating polynomials at specific values and determining if those values are roots.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is ƒ(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding zeros and evaluating the function at specific points.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i' (where i^2 = -1). In this context, evaluating the polynomial at a complex number like k = 1 - i requires understanding how to perform arithmetic operations with complex numbers, which is crucial for determining the value of ƒ(k).