Problem 113 - College Algebra

Question

Perform the indicated operations. $$ \left(x^{n}+2\right)\left(x^{n}-2\right)-\left(x^{n}-3\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(6x^n - 13\)

Step 1: Applying the Difference of Squares

Apply the difference of squares formula to the first part of the expression \((x^n+2)(x^n-2)\), which can be rewritten as \((x^n)^2 - (2)^2\) to get \(x^{2n} - 4\).

Step 2: Expanding the Square Term

Expand \((x^n - 3)^2\) using the expansion formula \(a^2 - 2ab + b^2\), replacing \(a\) with \(x^n\) and \(b\) with \(3\). The expanded term will then be \(x^{2n} - 6x^n + 9\).

Step 3: Subtracting the Expanded Terms

Subtract the expanded form of \((x^n - 3)^2\) from \(x^{2n} - 4\) which gives the final solution \(x^{2n} - 4 - x^{2n} + 6x^n - 9\).

Key Concepts

Difference of Squares

A fundamental concept in algebra is the 'difference of squares.' This term refers to an expression of the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). A concrete example would be when we have \(x^{2n} - 4\). Recognizing that 4 is a square number (specifically, \(2^2\)), we can rewrite our expression as \(x^{2n} - 2^2\).

When applying the difference of squares to the exercise \(\left(x^{n}+2\right)\left(x^{n}-2\right)\), the first term \(x^{n}\) gets squared, producing \(x^{2n}\), and the second term is also squared, yielding \(4\), resulting in the simplified form of \(x^{2n} - 4\). Understanding this principle allows us to simplify many algebraic expressions efficiently.

Key Takeaway

Always look for the characteristic \(a^2 - b^2\) structure in algebraic expressions. When we identify this pattern, the difference of squares formula provides a quick way to factor and simplify expressions.

Expanding Polynomials

Expanding polynomials is another essential operation in algebra. It involves taking something like \((x^n - 3)^2\) and transforming it into a polynomial where all terms are expressed without parentheses. To do this, we use the formula \(a^2 - 2ab + b^2\), which is derived from the FOIL method (First, Outer, Inner, Last).

In the context of the provided exercise, we set \(a = x^n\) and \(b = 3\). Thus, \(a^2\) becomes \(x^{2n}\), \(-2ab\) becomes \(-6x^n\), and \(b^2\) becomes \(9\). After applying this, we end up with the expanded form: \(x^{2n} - 6x^n + 9\).

Practical Insight

Expanding polynomials like this turns implicit multiplications into explicit additions and subtractions, preparing the expression for further simplifications, such as addition or subtraction with other polynomials.

Exponentiation Rules

Understanding exponentiation rules is vital when working with polynomials, especially when dealing with operations involving powers. In our exercise, we encounter expressions like \(x^{n}\) raised to a power. The rule for raising a power to a power is straightforward: you multiply the exponents.

For example, if we have \(x^n\) to be squared, as in the exercise \((x^n - 3)^2\), we apply the rule to the \(x^n\) term to get \(x^{2n}\). The exponentiation rules save us time and simplify the process of working with powers in algebraic expressions.

Core Principle

The exponentiation rules dictate that when we multiply powers with the same base, we add exponents, and when we raise a power to another power, we multiply the exponents. These rules are crucial when expanding polynomials or simplifying algebraic expressions.