Problem 41 - Beginning and Intermediate Algebra

Question

Factor completely. $$y^{2}-\frac{1}{25}$$

Step-by-Step Solution

Answer

\(y^{2} -\frac{1}{25} = (y - \frac{1}{5})(y + \frac{1}{5})\)

Step 1: Identify a and b

In the given expression, \(y^{2} -\frac{1}{25}\), let \(a = y\) and \(b = \frac{1}{5}\). Given expression can be rewritten as \(a^2 - b^2\).

Step 2: Apply the difference of squares formula

Apply the \(a^2 - b^2 = (a-b)(a+b)\) formula. Substitute the values of \(a\) and \(b\) that we identified in Step 1. \(y^{2} -\frac{1}{25} = (y - \frac{1}{5})(y + \frac{1}{5})\).

Step 3: Express the final answer

After applying the difference of squares formula and substituting the values of \(a\) and \(b\), the factored expression is: \((y - \frac{1}{5})(y + \frac{1}{5})\)

Key Concepts

Difference of Squares

The difference of squares is a special type of polynomial expression that can be easily factored once you understand its unique pattern. A difference of squares takes the form \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares. The key aspect here is the subtraction sign between the two perfect squares.To factor a difference of squares, you use the formula:

• \(a^2 - b^2 = (a - b)(a + b)\)

This formula allows you to split the expression into two binomials. For example, in the exercise \(y^2 - \frac{1}{25}\), \(y\) is our \(a\) and \(\frac{1}{5}\) is our \(b\). By applying the formula, we factor the expression into:

• \((y - \frac{1}{5})(y + \frac{1}{5})\)

Understanding and identifying the difference of squares is essential for factoring polynomials quickly and efficiently.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It involves solving equations and understanding how these symbols can represent numbers and relationships between them.In our context, algebra helps us understand how different factoring techniques work by manipulating expressions to simplify them. For instance, knowing algebra makes it easier to comprehend why terms like \(y^2\) and \(\frac{1}{25}\) can be adjusted to fit into a recognizable pattern like the difference of squares.In solving the original exercise, algebraic manipulation allowed us to identify that the expression \(y^{2} - \frac{1}{25}\) fits the classic pattern of the difference of squares. We applied a formula based on these algebraic principles to factor the polynomial efficiently.

Factoring Techniques

Factoring techniques are strategies used to rewrite expressions as the product of simpler expressions. Recognizing patterns like the difference of squares can greatly simplify the process of factoring.The main goal of factoring is to break down complex expressions into simpler components, making it easier to solve equations where these expressions appear. In algebra, factoring is crucial for simplifying expressions and solving polynomial equations.Here's a straightforward technique: identify the pattern in the polynomial first, then apply the appropriate formula. For example, when you see \(a^2 - b^2\), you immediately know you can use the difference of squares formula.By mastering this and other factoring techniques, you'll find it much easier to tackle a wide variety of algebraic problems, leaving room for more advanced mathematical concepts to be explored.