Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of 3 and $i , P ( 1 ) = - 4$
A) $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x + 3$
B) $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x - 3$
C) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 3$
D) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - x + 3$

Question

Quizplus · Accepted Answer

Since the polynomial has real coefficients and a complex zero $i$, its conjugate $-i$ must also be a zero. The zeros are 3, $i$, and $-i$. The polynomial can be formed by multiplying $(x-3)(x-i)(x+i)$, simplifying to $x^3 - 3x^2 + x - 3$. Checking $P(1) = 1 - 3 + 1 - 3 = -4$ confirms option B.

Find a Polynomial of Degree 3 with Real Coefficients That i,P(1)=−4i , P ( 1 ) = - 4i,P(1)=−4

Find a Polynomial of Degree 3 with Real Coefficients That $i , P ( 1 ) = - 4$