Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of $3 , i , - i$ and $P ( 2 ) = 25$
A) $P ( x ) = - 5 x ^ { 3 } + 15 x ^ { 2 } - 5 x + 15$
B) $P ( x ) = - 5 x ^ { 3 } + 15 x ^ { 2 } + 5 x - 15$
C) $P ( x ) = 5 x ^ { 3 } - 15 x ^ { 2 } - 5 x + 15$
D) $P ( x ) = 5 x ^ { 3 } - 15 x ^ { 2 } + 5 x - 15$

Question

Quizplus · Accepted Answer

Given the zeros $3, i, -i$, the polynomial can be expressed as $P(x) = k(x-3)(x-i)(x+i)$, where $k$ is a constant. Simplifying the factors gives $P(x) = k(x-3)(x^2+1)$. To find $k$, use the condition $P(2) = 25$: $25 = k(2-3)((2)^2+1)$, leading to $25 = -k(5)$, so $k = -5$. Thus, $P(x) = -5(x-3)(x^2+1)$, which simplifies to $P(x) = -5x^3 + 15x^2 - 5x + 15$, matching choice A.

Find a Polynomial of Degree 3 with Real Coefficients That 3,i,−i3 , i , - i3,i,−i

Find a Polynomial of Degree 3 with Real Coefficients That $3 , i , - i$