Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zeros of -5 and 1 - i, P(0)=10
A) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 x$
B) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 x + 10$
C) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 8 x + 10$
D) $P ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x + 5$

Question

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zeros of  -5  and  1 - i, P(0)=10 
A) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 x$
B) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 x + 10$
C) $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 8 x + 10$
D) $P ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x + 5$

Quizplus · Accepted Answer

Since the polynomial has real coefficients and a complex zero of $1 - i$, its conjugate $1 + i$ must also be a zero. The zeros are -5, $1 - i$, and $1 + i$. The polynomial can be formed by multiplying $(x + 5)(x - (1 - i))(x - (1 + i))$. Simplifying this product gives a polynomial, and adjusting the constant term to ensure $P(0) = 10$ leads to the correct polynomial $P(x) = x^3 + 3x^2 - 8x + 10$.

Find a Polynomial of Degree 3 with Real Coefficients That P(x)=x3+3x2−8xP ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 xP(x)=x3+3x2−8x

Find a Polynomial of Degree 3 with Real Coefficients That $P ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 8 x$