Problem 14 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$(3 x+5)^{2}-y^{2}$$

Step-by-Step Solution

Answer

\((3x + 5 - y)(3x + 5 + y)\)

Step 1: Recognize the Difference of Squares Formula

The difference of squares formula states that for any two expressions \(a\) and \(b\), the difference of their squares is given by \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Identify the Terms in the Expression

In the expression \((3x + 5)^2 - y^2\), observe that the expression is in the form \(a^2 - b^2\), where \(a = 3x + 5\) and \(b = y\).

Step 3: Apply the Difference of Squares Formula

Substitute \(a\) and \(b\) into the difference of squares pattern: \[(3x + 5)^2 - y^2 = (3x + 5 - y)(3x + 5 + y)\]

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler, multiplied components. This makes it easier for us to work with them, especially when solving equations. The difference of squares is one of the many techniques used to factor polynomials. It applies when an expression is in the form of two squared terms separated by a subtraction sign. To factor using this method:
\[a^2 - b^2 = (a - b)(a + b)\]

• Identify terms in the expression as perfect squares.
• Recognize the negative sign between the squares indicates we use "difference".
• Rewrite the expression using the difference of squares formula.

This technique simplifies complex algebraic expressions and aids in solving quadratic equations by providing a clear path to the solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In \((3x + 5)^2 - y^2\), you've got two distinct parts separated by subtraction. Understanding expressions as building blocks is vital. Here’s what each component represents:

- **Terms:** Individual parts like \(3x + 5\) and \(y\).- **Operators:** Signs such as \(+\) and \(-\) that determine how terms interact.- **Coefficients:** Numbers that multiply variables, e.g., 3 in \(3x\).
Recognizing specific patterns within expressions, like squares, helps in factoring and simplifying them. The difference of squares is just one method, demonstrating the importance of identifying key characteristics in expressions. It is essential because it allows for manipulations that simplify problems, making algebra more approachable. This practice builds a stronger foundation for tackling more complex equations.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, often expressed in the form \(ax^2 + bx + c = 0\). However, sometimes they disguise themselves in expressions like \((3x + 5)^2 - y^2\). By recognizing the structure as a difference of squares, we can factor them more effectively.
Using the difference of squares pattern:
\[(3x + 5)^2 - y^2 = (3x + 5 - y)(3x + 5 + y)\]
This reveals two simpler linear equations that strike at the heart of the quadratic process. Solving these linear equations can lead us to find the roots of the original equation, which are the solutions to the problem.

Here are steps to solve quadratic equations via factoring:

• Recognize the quadratic form and identify the pattern.
• Factor the equation using appropriate methods, such as the difference of squares.
• Solve the resulting linear equations to find solutions.

Understanding how to break apart quadratic equations into solvable components is crucial in algebra. It unlocks a path to finding the values of variables that make the equation true, which is the essence of algebraic problem-solving.