What is the remainder when the function f left parenthesis x right parenthesis equals x superscript 4 baseline minus 3 x cubed plus 7 x minus 1 is divided by left parenthesis x minus 2 right parenthesis?

Question

(1 point)
Responses

2
2

5
5

17
17

25

Answer 1

To find the remainder when the polynomial \( f(x) = x^4 - 3x^3 + 7x - 1 \) is divided by \( (x - 2) \), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \( f(x) \) when divided by \( (x - c) \) is \( f(c) \).

In this case, we need to evaluate \( f(2) \):

\[ f(2) = (2)^4 - 3(2)^3 + 7(2) - 1 \]

Calculating each term:

1. \( (2)^4 = 16 \)
2. \( -3(2)^3 = -3(8) = -24 \)
3. \( 7(2) = 14 \)
4. The final term is \( -1 \)

Now substituting back into the equation:

\[ f(2) = 16 - 24 + 14 - 1 \]

Calculating step by step:

\[ f(2) = 16 - 24 = -8 \] \[ -8 + 14 = 6 \] \[ 6 - 1 = 5 \]

Thus, the remainder when \( f(x) \) is divided by \( (x - 2) \) is \( \boxed{5} \).