Problem 109 - Intermediate Algebra

Question

Simplify each expression. $$ \frac{x^{32}-1}{x^{16}-1} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{16} + 1\).

Step 1: Recognize the Pattern

The numerator has the form of a difference of powers and can be rewritten as a difference of squares. Notice that \(x^{32} - 1 = (x^{16})^2 - 1^2\). This can be factored using the difference of squares.

Step 2: Apply Difference of Squares

Using the identity \(a^2 - b^2 = (a-b)(a+b)\), rewrite \(x^{32} - 1\) as \((x^{16} - 1)(x^{16} + 1)\). This gives us: \[\frac{(x^{16} - 1)(x^{16} + 1)}{x^{16} - 1}\]

Step 3: Cancel Common Factors

Notice that \(x^{16} - 1\) is a common factor in both the numerator and the denominator. Cancel out \(x^{16} - 1\) from the numerator and denominator. This leaves us with: \[x^{16} + 1\]

Step 4: Confirm Simplified Form

After cancelling the common factor, the expression is simplified fully. The simplified expression is \(x^{16} + 1\).

Key Concepts

Polynomial Simplification

Simplifying polynomials is an important skill in algebra that allows mathematical expressions to be more manageable and easier to work with. It involves reducing expressions by combining like terms, employing mathematical identities, and cancelling out factors when possible. In the case of difference of squares, we look for patterns where an expression can be rewritten using identities. For example, the difference of squares identity, \(a^2 - b^2 = (a-b)(a+b)\), helps in transforming and simplifying these expressions.

• Look for products or sums that have common factors that can be cancelled.
• Once common factors are removed, you achieve a simplified form that can often be easier to interpret.

By practicing polynomial simplification, students can improve their algebraic problem-solving skills, creating a strong foundation for more complex equations.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler elements called factors, whose product is the original expression. One powerful technique under factoring is the use of the difference of squares. For instance, consider an expression like \(x^{32} - 1\). Recognizing it as a difference of squares helps us to factor it efficiently into \((x^{16} - 1)(x^{16} + 1)\). Factoring is essential as it can lead to simplified expressions that are easier to evaluate or solve.
A few tips to factor effectively:

• Identify any common factors before applying more specific methods like difference of squares.
• Use identities and patterns, like quadratics, to see if parts of the expression can be expressed as differences of squares or other factorable forms.
• Always verify your factors by multiplying them back together to ensure they form the original expression.

Mastering factoring can not only help in simplifying expressions but also in solving equations and finding roots of polynomials.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. Understanding how to manipulate these through simplification and factoring is crucial. In algebra, recognizing patterns such as difference of squares can significantly reduce complexity and help in solving equations efficiently. Learning to navigate algebraic expressions involves:

• Identifying the structure of expressions to recognize where simplification can occur.
• Utilizing known identities or properties to transform expressions into simpler forms.
• Practicing the art of recognizing terms and factors within expressions that allow them to be broken down.

By refining your skills in interpreting and working with algebraic expressions, you create a toolkit that aids in problem-solving, investigating functions, and simplifying complex problems.