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) = x³ + 3x² -x + 1; k = 1+i

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Write down the coefficients of the polynomial ƒ(x) = x^3 + 3x^2 - x + 1. These are 1 (for x^3), 3 (for x^2), -1 (for x), and 1 (constant term).

Set up synthetic division using the given value k = 1 + i. Since k is a complex number, synthetic division will involve complex arithmetic.

Bring down the first coefficient (1) as it is. Then multiply this by k (1 + i) and add the result to the next coefficient (3). Continue this process for all coefficients:

For each step, multiply the current sum by k and add the next coefficient, carefully performing addition and multiplication with complex numbers.

After completing the synthetic division, the final number you get is the remainder, which equals ƒ(k). If this remainder is zero, then k is a zero of the polynomial; otherwise, it is not.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial and performing arithmetic operations to find the quotient and remainder quickly.

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 of a polynomial is a value k for which the polynomial evaluates to zero, meaning f(k) = 0. Determining whether k is a zero involves substituting k into the polynomial or using synthetic division to check if the remainder is zero.

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