Using the factors (5x + 4) and (x - 1), find the remaining factor(s) of $f ( x ) = - 20 x ^ { 4 } + 109 x ^ { 3 } - 30 x ^ { 2 } - 79 x + 20$ and write the polynomial in fully factored form.
A) $f ( x ) = ( 5 x + 4 ) ( - x + 5 ) ( 4 x - 1 ) ( x - 1 )$
B) $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( 4 x - 1 ) ( x + 1 )$
C) $f ( x ) = ( 5 x + 4 ) ( 5 x + 4 ) ( 4 x - 1 ) ( x - 1 )$
D) $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( x - 1 ) ^ { 2 }$
E) $f ( x ) = ( 5 x + 4 ) ( - x - 5 ) ^ { 2 } ( x + 1 )$

Question

Using the factors (5x + 4) and (x - 1), find the remaining factor(s) of $f ( x ) = - 20 x ^ { 4 } + 109 x ^ { 3 } - 30 x ^ { 2 } - 79 x + 20$ and write the polynomial in fully factored form.
A)  $f ( x ) = ( 5 x + 4 ) ( - x + 5 ) ( 4 x - 1 ) ( x - 1 )$
B)  $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( 4 x - 1 ) ( x + 1 )$
C)  $f ( x ) = ( 5 x + 4 ) ( 5 x + 4 ) ( 4 x - 1 ) ( x - 1 )$
D)  $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( x - 1 ) ^ { 2 }$
E)  $f ( x ) = ( 5 x + 4 ) ( - x - 5 ) ^ { 2 } ( x + 1 )$

Quizplus · Accepted Answer

Given the factors (5x + 4) and (x - 1), we can divide the polynomial $f(x) = -20x^4 + 109x^3 - 30x^2 - 79x + 20$ by these factors to find the remaining factors. The correct remaining factors, when combined with the given ones, result in the polynomial being fully factored as $f(x) = (5x + 4)(-x + 5)(4x - 1)(x - 1)$. This matches option A.

Using the Factors (5x + 4) and (X - 1) f(x)=−20x4+109x3−30x2−79x+20f ( x ) = - 20 x ^ { 4 } + 109 x ^ { 3 } - 30 x ^ { 2 } - 79 x + 20f(x)=−20x4+109x3−30x2−79x+20

Using the Factors (5x + 4) and (X - 1) $f ( x ) = - 20 x ^ { 4 } + 109 x ^ { 3 } - 30 x ^ { 2 } - 79 x + 20$