If f, of, x, equals, x, to the power 6 , minus, 4, x, cubed, plus, 2f(x)=x

Question

If f, of, x, equals, x, to the power 6 , minus, 4, x, cubed, plus, 2f(x)=x 
6
 −4x 
3
 +2, then what is the remainder when f, of, xf(x) is divided by x, plus, 1x+1?

GPT-4o mini · Answer

To find the remainder when $ f(x) = x^6 - 4x^3 + 2 $ is divided by $ x + 1 $, we can use the Remainder Theorem. This theorem states that the remainder of the division of a polynomial $ f(x) $ by $ x - a $ is $ f(a) $. In this case, we want to divide by $ x + 1 $, or $ x - (-1) $. Therefore, we will evaluate $ f(-1) $: $$ f(x) = x^6 - 4x^3 + 2 $$ Now, substituting $ -1 $ into $ f(x) $: $$ f(-1) = (-1)^6 - 4(-1)^3 + 2 $$ $$ = 1 - 4(-1) + 2 $$ $$ = 1 + 4 + 2 $$ $$ = 7 $$ Thus, the remainder when $ f(x) $ is divided by $ x + 1 $ is $ \boxed{7} $.