Question

Using the given zero, find one other zero of \( f(x) \). \(4 - 5i\) is a zero of \( f(x) = x^4 - 8x^3 + 42x^2 - 8x + 41\).

A) -4 - 5i
B) 1 - i
C) 4
D) 5i

Answer 1

This is the conjugate of the given zero 4−5i, as complex roots occur in conjugate pairs for polynomials with real coefficients.Therefore the correct answer is A) -4 - 5i.

Step-by-step explanation:

The given zero 4 - 5i implies that its conjugate, 4 + 5i, is also a zero for a polynomial with real coefficients. This is because complex roots occur in conjugate pairs for polynomials with real coefficients. Thus, the corresponding complex conjugate of 4 - 5i is 4 + 5i, making it another zero of the polynomial.

In this case, the correct answer is option A) -4 - 5i. It's important to recognize the conjugate relationship when dealing with complex roots in polynomials. The conjugate pair property is a consequence of the fact that if a + bi is a root, then a - bi must also be a root for polynomials with real coefficients. Therefore, by identifying the given zero and applying the conjugate relationship, we can determine the other zero as -4 - 5i.

In summary, the correct choice is A) -4 - 5i, and this is derived from the conjugate pair property of complex roots in polynomials with real coefficients, ensuring that both 4 - 5i and its conjugate 4 + 5i are zeros of the given polynomial.Therefore the correct answer is A) -4 - 5i.