Question

Factor the difference of two squares.\(x^{2}-\frac{1}{9}\)

Short answer

The factored form of \(x^{2}-\frac{1}{9}\) is \((x + \frac{1}{3})(x - \frac{1}{3})\).

Step-by-step solution

Identify a and b

Identify a and b where a and b are the square roots of the first and the second term respectively. Here, \(a = x\) and \(b = \frac{1}{3}\) because \(x^{2}\) is equivalent to \((x)^{2}\) and \(\frac{1}{9}\) is equivalent to \(\left(\frac{1}{3}\right)^{2}\).

Apply the Difference of Squares Formula

Next, apply the difference of squares formula, \(a^{2}-b^{2} = (a+b)(a-b)\). Substitute the values of a and b obtained in step 1 into this formula. That is, replace a with x and b with \(\frac{1}{3}\). So, the factored form of \(x^{2}-\frac{1}{9}\) becomes \((x + \frac{1}{3})(x - \frac{1}{3})\).

Key concepts

Factoring Polynomials

When we talk about factoring polynomials, we're referring to the process of breaking down a polynomial into simpler components called factors that, when multiplied together, give back the original polynomial. It's like finding the ingredients of a recipe that come together to make a delicious cake.

In algebra, one of the key recipes we use is the Difference of Two Squares method. This method applies when we have a polynomial that can be expressed as one square number subtracted from another square number. For example, in the exercise \(x^2 - \frac{1}{9}\), we see that \(x^2\) is the square of \(x\), and \(\frac{1}{9}\) is the square of \(\frac{1}{3}\).

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like \(x\)), and operations (such as addition and subtraction). In the context of the given exercise, \(x^2 - \frac{1}{9}\) is an algebraic expression that poses a relationship between \(x\) and a fraction. Understanding how to manipulate these expressions is fundamental in algebra, especially when simplifying or factoring.

To successfully factor an expression like the one in our example, it’s essential to recognize patterns and apply specific formulas, such as the Difference of Two Squares.

Factorization

Factorization is the action we take to decompose an algebraic expression into factors. Think of it as unpacking a box into the individual pieces needed to assemble a product. For example, the expression \(x^2 - \frac{1}{9}\) can be factored using the concept of the Difference of Two Squares, recognizing that each term is a perfect square, and then expressing the given expression as a product of binomials.

The formula \(a^2 - b^2 = (a + b)(a - b)\) guides us through this process. Picking out our \(a\) as \(x\) and our \(b\) as \(\frac{1}{3}\), we can rewrite our original expression as \((x + \frac{1}{3})(x - \frac{1}{3})\), showing the expression's factors clearly.

Quadratic Equations

Quadratic equations are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Factoring is a powerful method for solving these equations because it transforms the equation into a set of simpler ones, often linear. Once a quadratic is factored, the solutions can be found by setting each factor to zero.

While the provided exercise isn't a quadratic equation, the process of factoring polynomials is closely related to solving quadratics. If we had an equation like \(x^2 - \frac{1}{9} = 0\), factoring it into \((x + \frac{1}{3})(x - \frac{1}{3}) = 0\) would allow us to solve for \(x\) by applying the Zero Product Property, leading to the solutions \(x = \frac{1}{3}\) and \(x = -\frac{1}{3}\).