Question

If \( f(x) = 2x^{4} - 13x^{2} + ax + b \) is divisible by \( x^{2} - 3x + 2 \), then \( (a, b) \) is equal to

• A. (-9, -2)
• B. (6, 4)
• C. (9, 2)
• D. (2, 9)

Hint:

Remainder Theorem: If $(x-k)$ divides $f(x)$, then $f(k) = 0$.

Answer 1

The Correct Option is C

Step 1: Factorize Divisor
$x^2 - 3x + 2 = (x - 2)(x - 1)$. Since $f(x)$ is divisible by this, $f(1) = 0$ and $f(2) = 0$.
Step 2: First Equation
$f(2) = 2(2)^4 - 13(2)^2 + 2a + b = 32 - 52 + 2a + b = 0 \Rightarrow 2a + b = 20$.
Step 3: Second Equation
$f(1) = 2(1)^4 - 13(1)^2 + a + b = 2 - 13 + a + b = 0 \Rightarrow a + b = 11$.
Step 4: Solve System
Subtracting (ii) from (i): $(2a + b) - (a + b) = 20 - 11 \Rightarrow a = 9$. Substituting $a = 9$ into (ii): $9 + b = 11 \Rightarrow b = 2$.
Final Answer: (c)