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 + 3x + 4; k = 2+i

Verified step by step guidance

1. Identify the polynomial function and the value of \( k \). Here, \( f(x) = x^2 + 3x + 4 \) and \( k = 2 + i \), where \( i \) is the imaginary unit.
2. Set up synthetic division using \( k = 2 + i \) as the divisor. Write down the coefficients of \( f(x) \): 1 (for \( x^2 \)), 3 (for \( x \)), and 4 (constant term).
3. Begin synthetic division by bringing down the first coefficient (1) as is. Then multiply this by \( k = 2 + i \) and add the result to the next coefficient (3).
4. Continue the process: multiply the new sum by \( k = 2 + i \) and add it to the last coefficient (4). The final number you get is the remainder, which equals \( f(k) \).
5. If the remainder is zero, then \( k \) is a zero of the polynomial. If not, the remainder is the value of \( f(k) \), showing that \( k \) is not a zero.

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 long division process by using only the coefficients of the polynomial and the value k, making it faster to evaluate polynomials or find remainders.

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 = 2 + 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.