Problem 21 - Contemporary Precalculus

Question

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{4 / 5}-81$$

Step-by-Step Solution

Answer

Answer: The factored form of the expression is $$(x^{4 / 5} - 9)(x^{8 / 5} - 81)$$.

Step 1: Rewrite the expression as a difference of two squares

Rewrite the expression as a difference of two squares: $$x^{4 / 5} - 3^4$$, where $$a = x^{4 / 5}$$ and $$b = 3^2$$.

Step 2: Identify the variables 'a' and 'b'

Identify the variables 'a' and 'b': $$a = x^{4 / 5}$$ and $$b = 3^2 = 9$$.

Step 3: Apply the formula for factoring a difference of squares

Using the formula for the difference of squares, we have: $$(x^{4 / 5} - 9)((x^{4 / 5})^2 - (3^2)^2) = (x^{4 / 5} - 9)(x^{8 / 5} - 81)$$.

Step 4: Final answer

The factored expression is $$(x^{4 / 5} - 9)(x^{8 / 5} - 81)$$.

Key Concepts

Understanding the Difference of Squares

The difference of squares is an essential concept in algebra that expresses one number squared subtracted by another number squared. Formally, it is defined as \( a^2 - b^2 \). This can always be factored into the product of two binomials: \( (a - b)(a + b) \). This formula is crucial as it simplifies expressions and helps solve equations more efficiently.

In the given exercise, we applied the difference of squares on the expression \( x^{4/5} - 3^4 \). Here, \( x^{4/5} \) and \( 3^2 \) correspond to \( a^2 \) and \( b^2 \) respectively. Using the formula, the expression \( x^{4/5} - 3^4 \) can be rewritten as two factors: \( (x^{4/5} - 3) \) and \( (x^{4/5} + 3) \).

• Always identify \( a \) and \( b \), where \( a^2 \) and \( b^2 \) appear in the expression.
• Apply the factoring technique to transform the expression into the product of two binomials.

This technique simplifies algebraic manipulations and is used across polynomial expressions. Understanding this concept fortifies problem-solving skills in algebra.

What are Algebraic Expressions?

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They do not have an equality sign; hence they are different from equations. Algebraic expressions can be as simple as \( x + 2 \) or as complex as \( x^{4/5} - 81 \), which is the expression from the exercise.

Every algebraic expression is essentially a combination of constants and variables, linked by such operations like addition, subtraction, multiplication, and division. Here's the breakdown:

• Constants: Fixed numbers, like the '81' in \( x^{4/5} - 81 \).
• Variables: Symbols that represent numbers, such as \( x \).
• Exponents: Indicate how many times a number is multiplied by itself, like \( 4/5 \) in \( x^{4/5} \).

Understanding algebraic expressions is pivotal, as they form the foundation of algebraic problem-solving. Recognizing patterns in these expressions, such as the possibility to apply the difference of squares, hands you tools to factor and simplify them effectively.

Factoring Techniques Unveiled

Factoring is a method used to break down complex algebraic expressions into simpler, multipliable components. It plays a vital role in solving algebraic problems and simplifying expressions. One of the most common factoring techniques is using the difference of squares, which we applied in the given expression \( x^{4/5} - 3^4 \).

There are several factoring techniques you might encounter, including:

• Common Factor Extraction: Look for common factors in all terms of the expression. Extract them to simplify the expression.
• Difference of Squares: As discussed, utilize this when you have a subtraction of two perfect squares.
• Grouping: Used in expressions with four or more terms, where pairs of terms are factored separately and then combined.

Mastering these techniques allows you to tackle a wide range of problems more effectively. Especially when dealing with higher-degree polynomials and complex expressions, such skills become invaluable in both academic and real-world applications.