Problem 113 - Algebra and Trigonometry

Question

In Exercises 111–113, 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 result of performing the indicated operations is \(6x^{n} - 13\)

Step 1: Expand the Product of Binomials

First, apply the formula for the product of binomials to the first two terms. The formula is \((a+b)\)(a-b) = \(a^{2} - b^{2}\). Here, \(x^{n}\) is 'a' and 2 is 'b'. So, \((x^{n}+2)\) \((x^{n}-2)\) = \(x^{2n} - 4\)

Step 2: Expand the Binomial Square

Then, apply the formula for the square of binomials to the third term. The formula is \((a-b)^{2}\) = \(a^{2} - 2ab + b^{2}\). Here, \(x^{n}\) is 'a' and 3 is 'b'. So, \((x^{n}-3)^{2}\) = \(x^{2n}- 6x^{n} + 9\).

Step 3: Subtract Terms

Now, subtract the second expanded expression from the first using the formula: first - second. Hence, \(x^{2n} - 4\) - \((x^{2n} - 6x^{n} + 9)\) = \(6x^{n} - 13\)

Key Concepts

Binomial Expansion

Expanding binomials is a key algebraic skill used to simplify expressions. The binomial expansion involves formulas to manage expressions of two terms, multiplied or squared.

When multiplying binomials like \((a+b)(a-b)\), the difference of squares formula applies: \(a^{2} - b^{2}\). It's useful for expressions such as \((x^n + 2)(x^n - 2)\), where \(x^n\) and 2 are your binomial components.

For squaring binomials like \((a-b)^{2}\), use the square of a binomial formula: \(a^{2} - 2ab + b^{2}\). Here, that applies to \((x^n - 3)^2\). These expansions are critical for simplifying and solving complex algebraic expressions efficiently. The right use of formulas can make otherwise lengthy calculations quick and accurate.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. Understanding these expressions is essential for solving equations and performing algebraic operations.

Variables represent unknown values, allowing for general solutions to mathematical problems. Coefficients are numbers positioned in front of variables, scaling the variable's value.

Operations such as addition, subtraction, multiplication, and division link the components of an expression, determining how they interact and combine. In more complex expressions, exponentiation introduces powers, making elements like \(x^n\) possible.

Simplifying and manipulating these expressions underpins many mathematical processes, providing routes to solving equations and understanding numerical relationships.

Subtraction of Polynomials

Subtracting polynomials involves dealing with expressions that contain multiple terms. Each term consists of a coefficient and a variable raised to a power.

The subtraction process requires aligning like terms, which means matching terms with the same variable and exponent. Subtract the coefficients of these like terms directly. For instance, in the expression \(x^{2n} - 4\) subtracted by \((x^{2n} - 6x^n + 9)\), align and subtract like terms to simplify: the \(x^{2n}\) terms cancel out, as they equal zero when subtracted, leading to \(6x^n - 13\).

Understanding the process of subtracting polynomials helps in simplifying expressions and solving polynomial equations efficiently, providing clarity and simplification in larger algebraic problems.