Problem 45 - Algebra 2

Question

Factor each expression. $$ 9 x^{2}-1 $$

Step-by-Step Solution

Answer

\((3x + 1)(3x - 1)\)

Step 1: Identify the Type of Binomial

Observe that the given expression is a difference of squares since both 9 and 1 are perfect squares and the expression is in the form of \(a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

The formula for factoring a difference of squares is \(a^2 - b^2 = (a + b)(a - b)\). For the given expression, \(a^2 = 9x^2\) and \(b^2 = 1\), so we have \(a = 3x\) and \(b = 1\).

Step 3: Factor the Expression

Replace a and b with 3x and 1 in the formula to get the factors: \(9x^2 - 1 = (3x + 1)(3x - 1)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that allows us to simplify algebraic expressions and solve equations efficiently. It involves breaking down a polynomial into a product of simpler polynomials or binomials that, when multiplied together, give back the original polynomial. There are several common methods for factoring polynomials, including factoring out the greatest common factor, grouping, and using special factoring formulas like the difference of squares.

In the context of our exercise, the expression \( 9x^2-1 \) is a binomial, which presents a unique opportunity to apply one such special formula. It's important to recognize different forms polynomials can take, as it's not solely about applying rote techniques but understanding the underlying structure that directs us to the most efficient factoring method.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. Familiarity with perfect squares can be incredibly helpful in factoring polynomials, especially when dealing with the difference of squares method. In algebra, recognizing when a term is a perfect square can lead to quick and effective factorization.

The concept comes directly into play in the given exercise: both 9 and 1 are perfect squares since \(3^2 = 9\) and \(1^2 = 1\). This recognition is crucial because it cues us to the fact that the exercise can be approached through the lens of the difference of squares formula, making the factorization process straightforward.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation (raising to a power). These expressions can represent a wide range of mathematical relationships and are fundamental to understanding algebra.

For our problem, \( 9x^2-1 \) is an algebraic expression composed of terms that are perfect squares. The expression itself doesn't include arithmetic operations between different variables, which can often complicate the factoring process. Here, the simplicity of the expression's structure directly informs us of the strategy to employ for factorization: identifying the perfect square terms and applying the pertinent formula.

Binomial Expressions

Binomial expressions have exactly two terms. These terms may include variables, numbers, and exponents, but the key is that there are two distinct parts. In algebra, binomials are involved in a variety of operations, including addition, subtraction, and multiplication.

Our expression is a binomial, specifically, it is a difference of two perfect squares. In factoring such binomials, there is a very efficient method when the structure fits the pattern \(a^2-b^2\). In our textbook solution, identifying the expression as a binomial difference of squares is a clarifying first step, allowing for the application of the formula \(a^2-b^2=(a+b)(a-b)\) to find the factors quickly. This is a beautiful demonstration of how recognizing the type of expression can lead to a swift and elegant solution.