For the polynomial, one zero is given. Find all others.
-$$P ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25 ; 5$$
A) $1 + 2 i , 1 - 2 i$
B) $1 + \sqrt { 5 } , 1 - \sqrt { 5 }$
C) $- 1 + 2 \mathrm { i } , - 1 - 2 \mathrm { i }$
D) $1 + \sqrt { 5 } i , 1 - \sqrt { 5 } i$

Question

Quizplus · Accepted Answer

Given that 5 is a zero of the polynomial, we can use synthetic division or polynomial division to divide the polynomial by $x - 5$ to find the quotient. The quotient will be a quadratic polynomial, and its roots can be found using the quadratic formula or factoring if possible. The roots of the quadratic polynomial are the other zeros of the original polynomial. Since the coefficients of the original polynomial are real and one of the zeros is real (5), the remaining zeros must either be real or occur in complex conjugate pairs. The correct answer involves complex conjugate pairs, which is consistent with the Fundamental Theorem of Algebra and the nature of polynomials with real coefficients.

For the Polynomial, One Zero Is Given P(x)=x3−7x2+15x−25;5P ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25 ; 5P(x)=x3−7x2+15x−25;5

For the Polynomial, One Zero Is Given $P ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25 ; 5$