Problem 138 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\((x^{n} + 2)(x^{n} + 4)\

Step 1: Identifying basic form

Recognize that the given polynomial equation \(x^{2 n}+6 x^{n}+8\) matches the basic second degree polynomial form \(x^{2}+bx+c\), only with \(x^{n}\) instead of \(x\). So, we factor this equation similarly.

Step 2: Factoring the polynomial

We look for two numbers that multiply to \(c=8\) (the third term in the polynomial) and add to \(b=6\) (the coefficient of the second term). These two numbers are \(2\) and \(4\). So, the factored form of the polynomial will be: \(x^{n} + 2\) and \(x^{n} + 4\).

Step 3: Writing the final answer

Write down the final answer, which is the polynomial factored completely: \((x^{n} + 2)(x^{n} + 4).\)