Using the factors (x - 5) and (x + 2), find the remaining factor(s) of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x + 30$ and write the polynomial in fully factored form.
A) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x - 3 )$
B) $f ( x ) = ( x - 5 ) ( x + 2 ) ^ { 2 }$
C) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x + 3 )$
D) $f ( x ) = ( x - 5 ) ^ { 2 } ( x + 2 )$
E) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x + 1 )$

Question

Quizplus · Accepted Answer

Given the factors (x - 5) and (x + 2), we can perform polynomial division or use synthetic division to find the remaining factor. The polynomial $f(x) = x^3 - 6x^2 - x + 30$ can be factored completely by finding the roots that make the polynomial equal to zero. Since (x - 5) and (x + 2) are factors, we know that x = 5 and x = -2 are roots of the polynomial. To find the remaining factor, we can divide the polynomial by these factors, which will result in the remaining factor being (x - 3), making the fully factored form of the polynomial $f(x) = (x - 5)(x + 2)(x - 3)$.

Using the Factors (X - 5) and (X + 2) f(x)=x3−6x2−x+30f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x + 30f(x)=x3−6x2−x+30

Using the Factors (X - 5) and (X + 2) $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x + 30$