Question

Show that the polynomial f(x) = x⁴ + 4x² + 6 has no zeroes.

Answer 1

Let $t=x^2$ So, $f\left(t\right)=t^2+4t+6$

Now, to find the zeroes, we will equate$f\left(t\right)=0.$

$\Rightarrow t^2+4t+6=0$

$t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Now,$t=\frac{-4\pm \sqrt{16-24}}{2}$

$=\frac{-4\pm \sqrt{-8}}{2}$

Which is not a real number.

The zeroes of a polynomial should be real numbers.

∴The given f(x) has no zeroes.