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Chapter 5: Problem 54

Factor completely. \(x^{2}-\frac{1}{9}\)

Short Answer

Expert verified
(x - \frac{1}{3})(x + \frac{1}{3})

Step by step solution

01

- Recognize the form

Observe that the given expression is a difference of squares. The expression is in the form of \(a^2 - b^2\), where \(a = x\) and \(b = \frac{1}{3}\).
02

- Apply the difference of squares formula

The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). For our specific expression, substitute \(a = x\) and \(b = \frac{1}{3}\).
03

- Write the factored form

After substituting the values, the expression \(x^2 - \frac{1}{9}\) becomes \((x - \frac{1}{3})(x + \frac{1}{3})\). Therefore, the factored form of \(x^2 - \frac{1}{9}\) is \((x - \frac{1}{3})(x + \frac{1}{3})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of squares
A difference of squares is a specific type of algebraic expression. This occurs when you have two squared terms subtracted from one another. The general form is given by: \(a^2 - b^2\).
To identify a difference of squares, check for two conditions:
  • Both terms must be perfect squares.
  • There must be a subtraction operation between them.
The key insight is recognizing that such expressions can always be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).
This technique simplifies the expression, breaking it down into more manageable factors.
algebraic expressions
An algebraic expression is a combination of variables, numbers, and operations.
For example, \(x^2 - \frac{1}{9}\) is an algebraic expression. Breaking down these components is crucial:
  • \textbf{Variables:} Symbols like 'x' representing unknown values.
  • \textbf{Numbers:} Known constants, e.g., '1' or '9'.
  • \textbf{Operations:} Mathematical operations such as addition, subtraction, multiplication, and division.
Understanding algebraic expressions lays the groundwork for solving more complex math problems. When dealing with these expressions, follow clear steps like combining like terms and using factoring techniques.
factored form
The factored form of an algebraic expression is its representation as a product of its factors. Factoring helps simplify expressions and solve equations.
Take, for instance, our expression \(x^2 - \frac{1}{9}\). By recognizing it as a difference of squares, we applied the formula \(a^2 - b^2 = (a - b)(a + b)\).
  • Step 1: Identify the form \(a^2 - b^2\).
  • Step 2: Apply the formula, with \(a = x\) and \(b = \frac{1}{3}\).
  • Step 3: Write the factored form: \( (x - \frac{1}{3})(x + \frac{1}{3})\).
This process turned \(x^2 - \frac{1}{9}\) into its factored form \( (x - \frac{1}{3})(x + \frac{1}{3})\).
Factoring is a powerful tool in algebra for simplifying expressions and solving equations efficiently.

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