If f, of, x, equals, 2, x, to the power 5 , minus, 5, x, cubed, minus, 4f(x)=2x

Question

If f, of, x, equals, 2, x, to the power 5 , minus, 5, x, cubed, minus, 4f(x)=2x 
5
 −5x 
3
 −4, then what is the remainder when f, of, xf(x) is divided by x, minus, 1x−1?

GPT-4o mini · Answer

To find the remainder when $ f(x) = 2x^5 - 5x^3 - 4 $ is divided by $ x - 1 $, we can use the Remainder Theorem, which states that when a polynomial $ f(x) $ is divided by $ x - c $, the remainder is $ f(c) $. In this case, we are interested in the remainder when $ f(x) $ is divided by $ x - 1 $. So, we will evaluate $ f(1) $: $$ f(1) = 2(1)^5 - 5(1)^3 - 4 $$ Calculating this step by step: $$ = 2(1) - 5(1) - 4 $$ $$ = 2 - 5 - 4 $$ $$ = 2 - 9 $$ $$ = -7 $$ Therefore, the remainder when $ f(x) $ is divided by $ x - 1 $ is $$ \boxed{-7} $$