Question

If $x^2-3x+2$ is factor of $x^4-a{x}^2+b$, then

• A. $a=5$
• B. $b=4$
• C. all roots of $x^4-a{x}^2+b=0$ are real and distinct
• D. $a=4,b=5$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a=5$
B $b=4$
C all roots of $x^4-a{x}^2+b=0$ are real and distinct

Let $f\left(x\right)=x^4-a{x}^2+b$
As $x^2-3x+2=(x-2)(x-1)$ is a factor
$f\left(2\right)=2^4-a{2}^2+b=0\Rightarrow 16-4a+b=0\Rightarrow 4a-b=16$ ...(1)
$f\left(1\right)=1^4-a{1}^2+b=0\Rightarrow a-b=1$ ...(2)
Solving (1) and (2),we get
$a=5,b=4$
Hence, options 'A' and 'B' are correct.