Problem 128 - Algebra and Trigonometry

Question

Factor completely. $$x^{2 n}+6 x^{n}+8$$

Step-by-Step Solution

Answer

The completely factored form of the equation \(x^{2n} + 6x^n + 8\) is \((x^n + 2)(x^n + 4)\).

Step 1: Identify quadratic pattern

The quadratic expression is in the form \(x^{2n} + 6x^n + 8\). The pattern can be identified as a quadratic in \(x^n\), i.e., \(a(x^n)^2 + b(x^n) + c\), where \(a=1\), \(b=6\) and \(c=8\).

Step 2: Factor the quadratic

Next, factor the quadratic expression. This means finding two numbers that both add up to 6 (which is our \(b\) value) and multiply to 8 (our \(c\) value). With a bit of consideration, it is evident that the numbers 4 and 2 satisfy these conditions. 4 + 2 equals 6 and 4 times 2 equals 8.

Step 3: Write the factors

Finally, write the quadratic in factored form, replacing the \(x^n\) with the actual factors. The factored form of the equation will be \((x^n + 2)(x^n + 4)\).