Question

Choose the most appropriate options. 
Let \( f(x) = ax^3 + 5x^2 - bx + 1 \). If when divided by \( x - 1 \) it leaves a remainder of 5, and \( f(x) \) is divisible by \( 3x - 1 \), then

• A. \( a = 26, b = 10 \)
• B. \( a = 24, b = 12 \)
• C. \( a = 24, b = 10 \)
• D. \( a = 26, b = 12 \)

Hint:

Use the remainder theorem and divisibility rules to find the unknown coefficients.

Answer 1

The Correct Option is C

From the given information, we know that \( f(1) = 5 \) and the remainder theorem applies.
By substituting \( x = 1 \) into the polynomial and using the condition for divisibility by \( 3x - 1 \), 
we get the system of equations that gives us \( a = 24 \) and \( b = 10 \).