Write $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 25 x - 75$ as a product of linear factors.
A) $x = ( x - 3 ) ( x + 5 ) ^ { 2 }$
B) $x = ( x - 3 ) ^ { 2 } ( x - 5 i )$
C) $x = ( x - 3 ) ( x - 5 ) ^ { 2 }$
D) $x = ( x - 3 ) ( x + 5 i ) ( x - 5 i )$
E) $x = ( x + 3 ) ( x - 3 ) ( x + 5 )$

Question

Quizplus · Accepted Answer

To factor the given polynomial $f(x) = x^3 - 3x^2 + 25x - 75$, we first look for rational roots using the Rational Root Theorem. However, noticing that the polynomial does not factor easily into rational linear factors, we must consider the possibility of complex roots. Given the coefficients and the constant term, we can see that the polynomial is designed to have a real root and a pair of complex conjugate roots. The correct factorization that includes a pair of complex conjugate roots $(x + 5i)(x - 5i)$ and a real root is $f(x) = (x - 3)(x + 5i)(x - 5i)$, which corresponds to choice D. This factorization correctly represents the polynomial as a product of linear factors, including the necessary complex roots for the polynomial to be completely factored over the complex numbers.

Write f(x)=x3−3x2+25x−75f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 25 x - 75f(x)=x3−3x2+25x−75

Write $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 25 x - 75$