Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x² - 2x + 2; k = 1-i

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = x^2 - 2x + 2. These are 1 (for x^2), -2 (for x), and 2 (constant term).
2. Set up synthetic division using the coefficients [1, -2, 2] and the divisor k = 1 - i. Remember, synthetic division works with numbers, so treat k as a complex number.
3. Bring down the first coefficient (1) as it is. Then multiply this number by k = 1 - i, and write the result under the next coefficient.
4. Add the second coefficient (-2) and the product from the previous step. Write the sum below the line. Repeat the multiply and add process for the last coefficient.
5. The final number you get after the last addition is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial. If not, this remainder is the value of ƒ(k).

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - k). It simplifies the division process by using only the coefficients of the polynomial, making it faster and less error-prone than long division. It also helps evaluate the polynomial at k to check if k is a root.

Complex Numbers

Complex numbers include a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with i² = -1. Understanding how to perform arithmetic with complex numbers is essential when evaluating polynomials at complex values like k = 1 - i.

Zeros of a Polynomial

A zero (or root) of a polynomial is a value k for which the polynomial evaluates to zero, i.e., f(k) = 0. Determining whether a given number is a zero involves substituting it into the polynomial or using synthetic division to check if the remainder is zero.