When f(x)=x^4-5x^3+8x is divided by x-3, its remainder is -30. Which of the following is correct according to the Remainder Theorem?

A. f(-3)=30

B. f(-3)=-30

C. f(3)=-30

D. f(3)=30

1 answer

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder of this division is equal to \( f(c) \).

In your case, since \( f(x) = x^4 - 5x^3 + 8x \) is divided by \( x - 3 \) and the remainder is -30, we have:

\[ f(3) = -30 \]

Now let's examine the options:

A. \( f(-3) = 30 \) - This is not supported by the Remainder Theorem regarding \( x - 3 \).

B. \( f(-3) = -30 \) - We don't have information about \( f(-3) \) related to the division by \( x - 3 \).

C. \( f(3) = -30 \) - This is correct as per our findings.

D. \( f(3) = 30 \) - This contradicts our result \( f(3) = -30 \).

The only correct statement according to the Remainder Theorem in this case is:

C. f(3) = -30.