Question

given $f(x) = x^2 - 3x + k$, and the remainder when $f(x)$ is divided by $x - 9$ is 53, then what is the value of $k$?

Explanation:

Step1: Apply Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, we divide \( f(x) \) by \( x - 9 \), so \( a = 9 \), and the remainder is \( f(9) = 53 \).

Step2: Substitute \( x = 9 \) into \( f(x) \)

Substitute \( x = 9 \) into \( f(x)=x^{2}-3x + k \):
\[
f(9)=9^{2}-3\times9 + k
\]
Calculate \( 9^{2}=81 \) and \( 3\times9 = 27 \), so:
\[
f(9)=81 - 27 + k=54 + k
\]

Step3: Solve for \( k \)

We know \( f(9) = 53 \), so set up the equation:
\[
54 + k=53
\]
Subtract 54 from both sides:
\[
k=53 - 54=- 1
\]

Answer:

\( k=-1 \)