Problem 44 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$36 z^{2}-81 y^{4}$$

Step-by-Step Solution

Answer

The factored form is \(3(2z + 3y^2)(2z - 3y^2)\).

Step 1: Recognize the form

Identify that the expression \(36z^2 - 81y^4\) is a difference of squares. Recall that a difference of squares takes the form \(a^2 - b^2\).

Step 2: Write the expression in the form of difference of squares

Express \(36z^2\) and \(81y^4\) as perfect squares: \(36z^2 = (6z)^2\) and \(81y^4 = (9y^2)^2\). The expression becomes \((6z)^2 - (9y^2)^2\).

Step 3: Apply the difference of squares formula

The difference of squares formula is \(a^2 - b^2 = (a + b)(a - b)\). Apply this to \((6z)^2 - (9y^2)^2\): we have \((6z + 9y^2)(6z - 9y^2)\).

Step 4: Check for further factorization

Each term in \((6z + 9y^2)(6z - 9y^2)\) can be factored further. Factor out the greatest common factor: \(3(2z + 3y^2)(2z - 3y^2)\).

Key Concepts

Factoring

Factoring is a foundational concept in algebra that involves breaking down expressions into simpler components or 'factors'.
This is particularly useful for simplifying expressions or solving equations.In the context of the difference of squares, factoring involves recognizing and utilizing the structure of the expression to break it down. For instance, the expression given, \(36z^2 - 81y^4\), is identified as a difference of squares.

• The first step involves expressing each term as a square: \(36z^2\) becomes \((6z)^2\), and \(81y^4\) as \((9y^2)^2\).
• The formula for the difference of squares, \(a^2 - b^2 = (a + b)(a - b)\), then allows us to further factor the expression.
• The final result is a product of factors: \((6z + 9y^2)(6z - 9y^2)\).

Understanding factoring not only simplifies calculations but also provides insights into the properties and solutions of algebraic expressions.

Perfect Squares

A perfect square is an algebraic expression that is the square of a binomial. For example, when you see something like \((a + b)^2\), it expands to \(a^2 + 2ab + b^2\). In the context of the difference of squares, recognizing perfect squares is crucial to simplifying and factoring such expressions.In the original exercise of factoring \(36z^2 - 81y^4\), noting that both \(36z^2\) and \(81y^4\) are perfect squares is the key to the entire process.

• Breaking each down, \(36z^2\) is the square of \(6z\), and \(81y^4\) is the square of \(9y^2\).
• This recognition allows the use of the difference of squares formula easily, simplifying the expression to a product of basic factors: \((6z + 9y^2)(6z - 9y^2)\).

Grasping the concept of perfect squares not only helps in factoring but also in identifying potential solutions quickly in more complex algebraic tasks.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that stand in for patterns or quantities. They are the language of algebra and form the basis of how we solve equations and model real-world situations.The expression \(36z^2 - 81y^4\) we looked at is an example of an algebraic expression that can be simplified or manipulated using algebraic identities like the difference of squares formula.

• The process of factoring helps in reducing these expressions to their simplest algebraic pieces.
• Recognizing the structure and form, such as perfect squares in differences of squares, is integral to effectively handling algebraic expressions.

Understanding algebraic expressions and their manipulation is vital for progressing through algebraic concepts like solving equations and inequalities, working with functions, and even calculus.