Question

Is the following quadratic polynomial reducible or irreducible?
$f\left(x\right)=x^2-\sqrt{2}$

• A. Reducible with one real root
• B. Reducible with two real roots
• C. Irreducible
• D. None of these

Answer 1

Answer: (B)

Reducible with two real roots

Let $\sqrt{2}=a^2\Rightarrow a^2=2^{1/2}\Rightarrow a=2^{1/4}$

So $f\left(x\right)=x^2-a^2=(x+a)(x-a)=(x+2^{1/4})(x-2^{1/4})$, using $a^2-b^2=(a+b)(a-b)$

So given quadratic is reducible to two real roots