Question

The polynomial \(f(x)\) when divided by \((x-4)\) and \((x-6)\) leaves remainders 6 and 8 respectively. Then the remainder of \(f(x)\) when divided by \((x^2-10x+24)\) is:

• A. \(x+2\)
• B. \(2x+1\)
• C. \(-x+2\)
• D. Can't be found as \(\deg f(x)\) is not given

Hint:

When the divisor is quadratic, assume the remainder as \(mx+n\) and use the given remainder conditions to determine \(m\) and \(n\).

Answer 1

The Correct Option is A

Step 1: Write the divisor in factorized form. \[ x^2-10x+24=(x-4)(x-6) \] The remainder upon division by a quadratic divisor must be linear. Let the remainder be \[ R(x)=mx+n \]

Step 2: Use the given remainders. Since \[ f(4)=6 \] and \[ f(6)=8 \] the remainder must satisfy \[ R(4)=6 \] \[ 4m+n=6 \] \[ \cdots(1) \] and \[ R(6)=8 \] \[ 6m+n=8 \] \[ \cdots(2) \]

Step 3: Solve for \(m\) and \(n\). Subtracting, \[ 2m=2 \] \[ m=1 \] Substitute into (1), \[ 4+n=6 \] \[ n=2 \] Thus, \[ R(x)=x+2 \] \[ \boxed{x+2} \]