Question

Let $f\left(x\right)=x^2+bx+c$, where $b,c\in R$. If $f\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then the least value of $f\left(x\right)$, is

• A. $2$
• B. $3$
• C. $\frac{5}{2}$
• D. $4$

Answer 1

The correct option is D $4$

$f\left(x\right)$ is a factor of $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$.
Hence, $f\left(x\right)$ is also a factor of $3(x^4+6x^2+25)-(3x^4+4x^2+28x+5)$
Hence, $f\left(x\right)$ is also a factor of $14x^2-28x+70=14(x^2-2x+5)$.
Hence, $f\left(x\right)=x^2-2x+5$.
$f\left(x\right)=(x-1{)}^2+4$
To minimise $f\left(x\right)$, $x=1$
$f\left(1\right)=4$.
Hence, option D is correct.