Problem 137 - College Algebra Essentials

Question

$$\text { factor completely.}$$ $$x^{2 n}+6 x^{n}+8$$

Step-by-Step Solution

Answer

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

Step 1: Identify the format of the quadratic equation

The given equation \(x^{2 n}+6 x^{n}+8\) matches the standard form \(ax^2+bx+c\) where here \(a=1\), \(b=6\), \(x=x^n\), and \(c=8\).\nNote that, instead of \(x\), we're seeing \(x^n\). However, the factoring principle still applies.

Step 2: Factor the quadratic equation.

We need to find two numbers that multiply to \(c=8\) and add up to \(b=6\). The two numbers that satisfy this condition are 2 and 4. Thus, the quadratic equation can be rewritten as \(x^{2n}+2x^{n}+4x^{n}+8\). Next, factor by grouping: group the first two terms together and the last two terms together to get \((x^{2n}+2x^{n})+(4x^{n}+8)\). Now factor out the greatest common factor from each group. This gives \(x^n(x^n+2)+4(x^n+2)\). Now factoring out \((x^n+2)\) gives us \((x^n+2)(x^{n}+4)\). This is the factored form of the given quadratic equation.

Key Concepts

Algebra

Algebra is a fascinating branch of mathematics. It deals with symbols, numbers, and the rules for manipulating these symbols to solve equations and problems. One fundamental task in algebra is factoring, which is a method to break down complex expressions into simpler ones. This method can particularly help solve quadratic equations, where we might be looking at terms involving variables raised to the second power, such as \(x^2\).

In our exercise, we are working with an equation of the form \(x^{2n} + 6x^n + 8\). At a glance, it might seem more complicated due to the presence of the variable \(x\) in terms of \(x^n\). However, the foundational techniques used in algebra, like factoring quadratic equations, still apply. Recognizing the structure and applying algebraic techniques can simplify expressions significantly, making it easier to find solutions.

Polynomials

Polynomials are expressions that consist of variables and coefficients combined using operations such as addition, subtraction, multiplication, and non-negative integer exponents. An example would be \(x^{2n} + 6x^n + 8\), which is composed of three terms. The highest power of the variable in this polynomial is in \(x^{2n}\), leading to it being classified as a second-degree polynomial relative to \(x^n\).

Polynomials can vary in complexity, but understanding their structure helps in simplifying and solving them. In the polynomial given, importantly, we see a variable raised to another power, such as \(x^n\). This can still be handled like a normal quadratic—you just need to remember that \(x^n\) is our 'base' variable. The goal usually is to express the polynomial in a product of simpler polynomials, which can be further analyzed or solved.

Factoring by Grouping

Factoring by grouping is an invaluable tool, especially when dealing with polynomials that don't lend themselves easily to simple factoring methods. This technique involves splitting a polynomial into groups, where you can factor out common elements from each group separately and then combine them.

For instance, in the polynomial \(x^{2n} + 6x^n + 8\), you start by re-writing it in such a way that helps to group terms together: \((x^{2n}+2x^n) + (4x^n+8)\). Then, you factor out the greatest common factor from each group: from \((x^{2n}+2x^n)\), take out \(x^n\), leaving \(x^n(x^n + 2)\), and from \((4x^n+8)\), take out \(4\), leaving \(4(x^n+2)\).

Finally, you combine these into \((x^n+2)(x^n+4)\). This stepwise approach to grouping not only makes it easier to handle the polynomial but also provides insights into its structure and solutions.