Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of 4 and i, $\mathrm { P } ( 1 ) = - 6$
A) $P ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 4$
B) $P ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x + 4$
C) $P ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x + 4$
D) $P ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 4$

Question

Quizplus · Accepted Answer

The polynomial must have zeros at 4, i, and -i (since complex roots come in conjugate pairs). This gives the factors (x - 4), (x - i), and (x + i). The polynomial is $P(x) = (x - 4)(x^2 + 1)$. Expanding this gives $P(x) = x^3 - 4x^2 + x - 4$. Checking $P(1) = -6$ confirms the correct polynomial.

Find a Polynomial of Degree 3 with Real Coefficients That P(1)=−6\mathrm { P } ( 1 ) = - 6P(1)=−6

Find a Polynomial of Degree 3 with Real Coefficients That $\mathrm { P } ( 1 ) = - 6$