Problem 45 - Algebra and Trigonometry

Question

Factor the difference of two squares. $$x^{4}-16$$

Step-by-Step Solution

Answer

\((x - 2)(x + 2)(x^2 + 4)\

Step 1: Recognize the Difference of Two Squares

The first step is to recognize that \( x^4 - 16 \) can be written as a difference of two squares. \(x^4\) can be re-written as \((x^2)^2\) and 16 as \(4^2\). So, \( x^4 - 16 \) becomes \( (x^2)^2 - (4)^2 \).

Step 2: Apply the Difference of Two Squares Formula

Next, apply the difference of two squares formula \( a^2 - b^2 = (a - b)(a + b) \), where a is \(x^2\) and b is 4. This gives \( (x^2-4)(x²+4) \) as the factorized form.

Step 3: Factorize Further if Possible

After the initial factorization, observe whether the factors themselves are differences of squares, and if so, repeat the process. Here, \(x^2 -4\) can be factorized further using the same principle to \( (x - 2)(x + 2) \). The factor \(x^2 + 4\), however, cannot be factorized in real numbers so it stays as is. Hence, the fully factorized form of the given expression is \((x - 2)(x + 2)(x^2 + 4)\).