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:

If polynomial divisible by \((x-r_1)(x-r_2)\), then \(f(r_1)=0\) and \(f(r_2)=0\). Use these to form equations for unknowns.

Answer 1

The Correct Option is C

Step 1: Factor divisor.
\[ x^2-3x+2=(x-1)(x-2) \]
Step 2: Use factor theorem.
If divisible, then:
\[ f(1)=0,\quad f(2)=0 \]
Step 3: Apply \(f(1)=0\).
\[ f(1)=2(1)^4-13(1)^2+a(1)+b =2-13+a+b \]
\[ a+b-11=0 \Rightarrow a+b=11 \]
Step 4: Apply \(f(2)=0\).
\[ f(2)=2(16)-13(4)+2a+b \]
\[ =32-52+2a+b \]
\[ 2a+b-20=0 \Rightarrow 2a+b=20 \]
Step 5: Solve simultaneous equations.
\[ a+b=11 \]
\[ 2a+b=20 \]
Subtract:
\[ a=9 \]
Then:
\[ b=11-9=2 \]
Final Answer:
\[ \boxed{(9,2)} \]