Question

If $\alpha$ is a zero of the polynomial $p\left(x\right)=x^4+x^2-1$, then the value of the expression ${\alpha }^6+{2\alpha }^4$ can not be

• A. rational
• B. neither prime nor composite
• C. negative integer
• D. complex number

Answer 1

Answer: (C)

negative integer

$p\left(x\right)=x^4+x^2-1$

since is a zero of p(x)

so , ${\alpha }^4+{\alpha }^2-1=0$

$\Rightarrow {\alpha }^4+{\alpha }^2=1$

Now ,

${\alpha }^6+2{\alpha }^4$

$={\alpha }^2\left({\alpha }^4+2{\alpha }^2\right)$

$={\alpha }^2\left({\alpha }^4+{\alpha }^2+{\alpha }^2\right)$

$={\alpha }^2\left(1+{\alpha }^2\right)\ge 0$

so, value of ${\alpha }^6+2{\alpha }^4$ cannot be a negative integer