Question

If $x^2-3x+2$ is a factor of $x^4-p{x}^2+q$, then the value of $p$ and $q$ are

• A. $5,-4$
• B. $5,4$
• C. $-5,4$
• D. $-5,-4$

Answer 1

Answer: (B)

$5,4$

Factors of $x^2-3x+2$ are $(x-1)$ and $(x-2)$.

Let $f\left(x\right)=x^4-p{x}^2+q$

Since, $f\left(x\right)$ is divisible by $(x-1)$ and $(x-2)$

$\therefore f\left(1\right)=0$ and $f\left(2\right)=0$

$\Rightarrow f\left(1\right)=1^4-p(1{)}^2+q=0$

and $f\left(2\right)=2^4-p(2{)}^2+q=0$

$1-p+q=0....\left(i\right)$

and $16-4p+q=0....\left(ii\right)$

$\left(i\right)-\left(ii\right)$, we get
$-15+3p=0\Rightarrow p=5$

putting value of $p$ in eq. $\left(i\right)$, we get $q=4$

Thus, we get
$p=5$ and $q=4$.