Problem 79 - Algebra 2

Question

Factor each expression. $$ 25 z^{2}-9 $$

Step-by-Step Solution

Answer

\((5z + 3)(5z - 3)\)

Step 1: Recognize The Difference of Squares Pattern

Identify that the given expression is in the form of a difference of squares, which is an algebraic pattern that can be factored. The difference of squares pattern is given by: \(a^2 - b^2 = (a + b)(a - b)\).In this case, the expression \(25z^2 - 9\) can be viewed as \((5z)^2 - (3)^2\).

Step 2: Apply the Difference of Squares Formula

Using the formula for the difference of squares, we factor the expression using the values identified for \(a\) and \(b\). Here, \(a = 5z\) and \(b = 3\), so we apply the formula to get:\((5z + 3)(5z - 3)\).

Key Concepts

Algebraic Patterns

Algebraic patterns are recurring themes or strategies used to solve mathematics problems, specifically within the realm of algebra. Understanding these patterns can greatly simplify complex problems. For instance, recognizing a pattern in a sequence can tell us how to find the nth term, or noticing a particular form can indicate a specific method of factoring.

When we look at any algebraic expression, identifying specific characteristics or patterns is key. Recognizing a pattern involves seeing the underlying structure of the expression. For example, when we notice terms are all raised to the same power, we might consider if they fit the pattern of a square of a binomial or the difference of squares. Patterns are like the backbone of algebra; they guide us in identifying the next steps in solving an equation or simplifying an expression.

Difference of Squares

The difference of squares is a specific algebraic pattern where two perfect squares are subtracted from one another. These can be factored into a product of binomials that have the same terms but opposite signs. The general formula for the difference of squares is \[a^2 - b^2 = (a + b)(a - b)\].

This pattern emerges because when the binomials \(a + b)\) and \(a - b)\) are multiplied together, the middle terms cancel out, leaving only the difference of two squares. Recognizing this pattern is essential because it allows us to transform a seemingly complex expression into a more manageable form. This method not only simplifies the expression but also helps in solving equations where such a pattern is present.

Factoring Expressions

Factoring expressions is the process of breaking down complicated algebraic expressions into simpler, multipliable factors or products. It greatly aids in solving equations, simplifying expressions, and finding zeroes of functions. Factoring can be thought of as reverse multiplication; it is the process of finding the elements that were multiplied together to get the original expression.

There are several factoring techniques, such as pulling out the greatest common factor (GCF), grouping, using the difference of squares, factoring perfect square trinomials, and other special products. The choice of method depends on the form of the expression. Successful factoring requires careful observation and recognition of potential patterns and relationships between the terms in the expression.

Algebra 2

Algebra 2 is an advanced course in mathematics that builds upon the concepts learned in Algebra 1. It takes students deeper into the world of algebraic functions, equations, and expressions. In Algebra 2, students encounter polynomials, complex numbers, exponential and logarithmic functions, and more intricate systems of equations, among other topics.

Skills developed in Algebra 2, such as factoring complex expressions and solving higher-degree equations, are pivotal in mathematics and sciences. Problems like factoring the difference of squares are typical exercises in this course, demonstrating how more advanced algebra prepares students to solve real-world problems. An understanding and mastery of these principles are not just academic exercises; they are tools for later courses in mathematics and applied sciences.