Problem 25 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ w^{2}-9 y^{2} $$

Step-by-Step Solution

Answer

The factored expression of \(w^2-9y^2\) is \((w-3y)(w+3y)\)

Step 1: Identify the components

Identify \(a\) and \(b\) from our expression \(w^2-9y^2\). Here, \(a\) is \(w\) and \(b\) is \(3y\) because \(3y\) squared equals to \(9y^2\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) to factor the expression. Substituting \(w\) for \(a\) and \(3y\) for \(b\) we get: \(w^2 - 9y^2 = (w-3y)(w+3y)\)

Step 3: Check your solution

To make sure the solution is correct, expand the factored form to confirm it matches the original expression. Multiply \((w-3y)\) by \((w+3y)\), which equals to \(w^2-9y^2\), verifying our solution.

Key Concepts

Difference of Squares

Understanding the concept of the "difference of squares" is essential in factoring algebraic expressions. The difference of squares refers to a special case in algebra where an expression is structured as the subtraction of two squared terms. Imagine you see something like \( a^2 - b^2 \). This can be rewritten in a factored form using the formula:

• \(a^2 - b^2 = (a-b)(a+b)\)

What makes this approach unique is that it applies exclusively to expressions that comprise two perfect squares separated by a minus sign. Such expressions are notably straightforward to factor once identified.
For example, the exercise you saw \( w^2 - 9y^2 \) perfectly fits the criteria. Here, \( w^2 \) and \( 9y^2 \) are both perfect squares, and they have a minus sign between them. Hence, applying the difference of squares formula is straightforward and efficient.

Factoring Techniques

Factoring techniques are various methods used to simplify or rearrange polynomial expressions to make them easier to work with or evaluate. Understanding these methods is key to mastering algebra. One prominent technique is identifying patterns, such as the difference of squares, as described earlier. Other factoring techniques include:

• **Factoring by grouping:** Useful when dealing with four-term polynomials. Group terms to factor out the greatest common factor from each pair and see if a common factor emerges.
• **Factor trinomials:** Typically expressed in the form \( ax^2 + bx + c \). The trick is to find two numbers that multiply to \( ac \) and add to \( b \).

While the techniques vary, the primary aim remains the same: to express the polynomial as a product of simpler factors, which is crucial for solving equations, graphing, and simplifying expressions further.

Quadratic Expressions

Quadratic expressions are one of the most common forms you'll encounter in algebra. They are polynomial expressions of degree 2, generally taking the form \( ax^2 + bx + c \). These can be seen in many practical scenarios ranging from physics problems to even business calculations. When tackling quadratic expressions, factoring is a powerful tool to simplify or solve them with ease. Here's why:

• Factoring allows you to express the quadratic in a simpler form, which is crucial for solving equations by setting each factor to zero, known as the Zero Product Property.
• It also helps in identifying the roots or solutions of the quadratic equation, represented graphically as the points where the parabola intersects the x-axis.

For instance, recognizing that the expression \( w^2 - 9y^2 \) is indeed a quadratic one allows you to apply specialized techniques like the difference of squares for efficient factoring. This insight simplifies both solving and analyzing the expression in mathematical contexts.