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?

Answer 1

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 \( \boxed{7} \).