Question

the expression ( x^3 - kx^2 + 2x + 5 ) leaves a remainder of 2 when divided by ( x - 3 ). find 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, \( f(x)=x^{3}-kx^{2}+2x + 5 \) and \( a = 3 \) (since we divide by \( x-3 \)), and the remainder is 2. So we set up the equation \( f(3)=2 \).

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=3^{3}-k\times3^{2}+2\times3 + 5 \)
\( f(3)=27-9k + 6+5 \)
\( f(3)=38 - 9k \)

Step2: Solve for k

Since the remainder is 2, we have the equation:
\( 38-9k=2 \)

Subtract 38 from both sides:
\( - 9k=2 - 38 \)
\( -9k=-36 \)

Divide both sides by - 9:
\( k=\frac{-36}{-9}=4 \)

Answer:

4