Problem 46 - Algebra and Trigonometry

Question

Factor the difference of two squares. $$x^{4}-1$$

Step-by-Step Solution

Answer

The factored form of \(x^{4}-1\) is \((x^{2}+1)(x+1)(x-1)\).

Step 1: Identify the difference of two squares

The given expression is \(x^{4}-1\). Note that \(x^{4}\) can be written as \((x^{2})^{2}\) and 1 as \(1^{2}\). So the expression is in the form \(a^{2} - b^{2}\).

Step 2: Apply the formula

Using the formula \(a^{2} - b^{2} = (a+b)(a-b)\), substitute \(a = x^{2}\) and \(b = 1\). By doing this, the expression \(x^{4}-1\) can be re-written to \((x^{2}+1)(x^{2}-1)\).

Step 3: Refactor the second difference of squares

The term \(x^{2}-1\) can further be factored due to being a difference of squares. We have \(a = x\) and \(b = 1\), so the expression becomes \(x^{2}-1 = (x+1)(x-1)\).

Step 4: Rewrite with all factors

Putting it all together, the expression \(x^{4}-1\) has been factored into \((x^{2}+1)(x+1)(x-1)\). This is the fully factored form of the expression.