How to Factor x⁴ - 2: A Step‑By‑Step Guide
Factoring polynomial expressions is a fundamental skill in algebra that opens the door to solving equations, simplifying functions, and understanding the behavior of mathematical models. When faced with an expression like x⁴ - 2, many students instinctively reach for the difference of squares formula, only to realize that the terms do not immediately fit the pattern *a²
That said, the expression can still be broken down by treating it as a difference of two perfect squares. Notice that
[ x^{4}= (x^{2})^{2}\qquad\text{and}\qquad 2 = (\sqrt{2})^{2}. ]
Thus
[ x^{4}-2 = (x^{2})^{2}-(\sqrt{2})^{2}. ]
Applying the identity (a^{2}-b^{2}=(a-b)(a+b)) gives
[ x^{4}-2 = (x^{2}-\sqrt{2})(,x^{2}+\sqrt{2},). ]
Each factor is a quadratic that can be examined separately.
Real‑valued factorisation
The first quadratic, (x^{2}-\sqrt{2}), has real roots because
[ x^{2}-\sqrt{2}=0 ;\Longrightarrow; x=\pm\sqrt[4]{2}. ]
Hence
[ x^{2}-\sqrt{2}= (x-\sqrt[4]{2})(x+\sqrt[4]{2}). ]
The second quadratic, (x^{2}+\sqrt{2}), has no real zeros (its discriminant is negative), so it remains irreducible over the real numbers.
Complex‑valued factorisation
If complex numbers are permitted, the second quadratic can be split as well:
[ x^{2}+\sqrt{2}=0 ;\Longrightarrow; x=\pm i\sqrt[4]{2}, ]
which yields
[ x^{2}+\sqrt{2}= (x
