Question

Factorize the equation $x^4 - 1 = 0$.

Answer 1

Hint: Since the given equation is bi-quadratic (degree 4), we will convert it into a quadratic equation, and then solve the roots using splitting the middle term method.

Complete step-by-step solution -
The given equation is $x^4 - 1 = 0$
To convert it into a quadratic equation, let us assume $x^2 = t$. So we can write that-
$t^2 - 1 = 0$
Using the property-
$a^2 - b^2 = (a - b)(a + b)$
$(t - 1)(t + 1) = 0$
We assumed that $t = x^2$, so we can write that-
$(x^2 - 1)(x^2 + 1) = 0$
Using the property again, we can write that-
$(x^2 + 1)(x - 1)(x + 1) = 0$

This is the factorization of the given equation.

Note: Often a common mistake is that instead of factoring, students find the roots of the equation, which is wrong. Also, we should write all the possible solutions and then eliminate them as and when required.