Question

If $f\left(x\right)$ is a polynomial of degree $5$ with real coefficients such that $f\left(|x|\right)=0$ has $8$ real roots, then $f\left(x\right)=0$ has

• A. $4$ real roots
• B. $5$ real roots
• C. $3$ real roots
• D. $nothingcanbesaid$

Answer 1

The correct option is B $5$ real roots

Given, $f\left(|x|\right)=0$ has $8$ real roots
$\Rightarrow f\left(\sqrt{x^2}\right)=0$ has $8$ real roots
$\Rightarrow f\left(x\right)=0$ has $4$ positive roots.
Since, $f\left(x\right)$ is a polynomial of odd degree.
So,$f\left(x\right)$ cannot have even number of real roots .
And since, total number of roots are $5$, so one root will be negative.
So, $f\left(x\right)=0$ has all five roots real.