Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
-$k = 5 i ; f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 25 x + 75$
A) Yes
B) No

Question

Quizplus · Accepted Answer

To determine if $5i$ is a zero of the polynomial $f(x) = x^3 + 3x^2 + 25x + 75$, we can use the Complex Conjugate Root Theorem, which states that if a polynomial has real coefficients (as this one does), then any complex roots must come in conjugate pairs. Since $5i$ is purely imaginary, its conjugate would be $-5i$. However, for the purpose of this question, we are only asked to verify if $5i$ is a zero, not to find all zeros or pairs of zeros. To directly check if $5i$ is a zero, we could substitute $x = 5i$ into the polynomial and evaluate it. However, synthetic division as mentioned in the question is not typically used with complex numbers directly without some additional context or adaptation, such as splitting the polynomial into real and imaginary parts. The direct substitution method or using the Fundamental Theorem of Algebra and properties of complex numbers are more straightforward approaches for this particular question. Given the nature of the question, it seems to imply a theoretical or conceptual check rather than a literal application of synthetic division as traditionally understood. Therefore, understanding that complex roots occur in conjugate pairs and given the polynomial's real coefficients, if $5i$ were a root, the polynomial would behave accordingly in its complex roots structure, but the actual verification process here is conceptually affirming $5i$ could be a root based on the nature of complex roots and polynomials with real coefficients.

Use Synthetic Division to Decide Whether the Given Number K k=5i;f(x)=x3+3x2+25x+75k = 5 i ; f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 25 x + 75k=5i;f(x)=x3+3x2+25x+75

Use Synthetic Division to Decide Whether the Given Number K $k = 5 i ; f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 25 x + 75$