Question

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

Answer 1

Let t = x² 

So, f(t) = t² + 4t + 6 

Now, to find the zeroes, we will equate f(t) = 0 

⇒ t² + 4t + 6 = 0

Now, t = \(\frac{-4±\sqrt{16-24}}2\)

= \(\frac{-4±\sqrt{-8}}2\)

= -2 ± \(\sqrt{-2}\)

i.e., x² = -2 ± \(\sqrt{-2}\)

⇒ x = \(\sqrt{-2±\sqrt{-2}}\) which is not a real number. 

The zeroes of a polynomial should be real numbers. 

∴The given f(x) has no zeroes.