Problem 12 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$16-49 y^{2}$$

Step-by-Step Solution

Answer

The factored form of \(16 - 49y^2\) is \((4 - 7y)(4 + 7y)\).

Step 1: Identify a and b

Looking at the expression \(16 - 49y^2\), our \(a\) and \(b\) can be identified as \(a = 4\) (since \(4^2 = 16\)) and \(b = 7y\) (since \((7y)^2 = 49y^2\)).

Step 2: Apply the Difference of Squares Formula

Once we have identified \(a\) and \(b\), we can now apply the formula for the difference of two squares which is \((a - b)(a + b)\). So, we replace \(a\) with \(4\) and \(b\) with \(7y\) to have \((4 - 7y)(4 + 7y)\).

Key Concepts

Factoring Techniques

Factoring techniques are essential tools in algebra that make equations easier to work with. One of the most common techniques is factoring using special formulas, like the Difference of Squares. This particular technique is used when you have a subtraction between two squared terms.
Factorization essentially involves rewriting an expression as a product of its factors. This is helpful in simplifying expressions and solving equations. In our example with the expression \(16 - 49y^2\), we use the difference of squares method.

• Identify the structure: Determine if the expression is in the form \(a^2 - b^2\).
• Apply the formula: Recognize that \( a^2 - b^2 = (a - b)(a + b) \).

Simple rules like these allow you to transform complex expressions into more manageable forms. Understanding these techniques enhances problem-solving skills and provides a foundation for more advanced algebraic manipulation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can include terms like \(49y^2\) and constants like \(16\). An algebraic expression doesn't have an equality sign like an equation does.
When dealing with expressions such as \(16 - 49y^2\), it is crucial to accurately interpret each component. In this case, we identify two square terms: \(16\) and \(49y^2\). The process of working with algebraic expressions often involves rewriting them to reveal their underlying structure.
Here is how you handle an expression:

• Identify terms and operations involved (addition, subtraction, etc.).
• Recognize special patterns or identities, like the difference of squares.
• Simplify by applying appropriate algebraic properties and formulas.

Grasping the nature of algebraic expressions is essential for anyone navigating the world of mathematics, as these form the backbone of algebraic thought.

Mathematical Formulas

Mathematical formulas are critical in unraveling complex problems. They serve as the tools that link different components of an expression or equation together. In algebra, formulas like the difference of squares provide a systematic way to simplify expressions.
The difference of squares formula is specifically useful when you have two terms, both squared, and separated by a subtraction. The formula is:

• \(a^2 - b^2 = (a - b)(a + b)\)
• This pattern helps in simplifying expressions by revealing their factors.

For the expression \(16 - 49y^2\), identifying \(a = 4\) and \(b = 7y\) allows us to apply the formula directly. This results in \((4 - 7y)(4 + 7y)\), turning a more complex subtraction into two simpler factors with a clear structure.
Understanding these formulas not only simplifies calculations but also unveils deeper insights into the nature of algebraic relationships.