Problem 77 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$x^{2}+64$$

Step-by-Step Solution

Answer

The polynomial \(x^{2}+64\) is prime over real numbers but can be factored over the set of complex numbers as \((x + 8i)(x - 8i)\).

Step 1: Identify the form of the expression

The given expression \(x^{2}+64\) is of the form \(a^{2} + b^{2}\), which represents a sum of squares, i.e. \(a^{2} + b^{2} = (a+bi)(a-bi)\) in complex numbers.

Step 2: Apply the sum of squares factoring rule using imaginary numbers

As mentioned in step 1, we can use the formula \(a^{2} + b^{2} = (a+bi)(a-bi)\) to factor the given expression. Here, a is \(x\) and b is \(8\) since because \(\sqrt{64} = 8\). So, \(x^{2} + 64\) can be factored over the set of complex numbers as \((x + 8i)(x - 8i)\).