Problem 52 - Algebra 2

Question

Factor each expression completely. $$ 18 z^{2}-8 $$

Step-by-Step Solution

Answer

The expression \(18z^2-8\) factors completely to \(2(3z+2)(3z-2)\).

Step 1: Recognize the common factor

First, identify if there is a common factor that can be factored out from both terms. In this case, both 18 and 8 can be divided by 2.

Step 2: Factor out the common factor

Factor out the common factor of 2 from both terms. This will give us: \(2(9z^2 - 4)\).

Step 3: Identify the Difference of Squares

Recognize that the expression inside the parentheses, \(9z^2 - 4\), is a difference of squares since \(9z^2 = (3z)^2\) and \(4 = 2^2\).

Step 4: Apply the Difference of Squares Formula

Use the difference of squares formula, which is \(a^2 - b^2 = (a + b)(a - b)\), to factor \(9z^2 - 4\). Here, \(a = 3z\) and \(b = 2\), so the expression becomes \((3z + 2)(3z - 2)\).

Step 5: Combine the Factored Terms

Combine the outside factor of 2 with the factored form of the parentheses to get the completely factored expression: \(2(3z + 2)(3z - 2)\).

Key Concepts

Difference of Squares

The difference of squares is a specific type of factoring that students encounter in algebra. It refers to expressions of the form \(a^2 - b^2\), which can be factored into \(a + b)(a - b)\). The key to recognizing a difference of squares is to look for two terms that are both perfect squares, separated by a subtraction sign.

Let's illustrate this concept with an example from the provided exercise. The expression within the parentheses, after factoring out the common factor, is \(9z^2 - 4\). Here, \(9z^2\) is a perfect square because it’s the square of \(3z\): \(3z)^2\). Similarly, \(4\) is a perfect square of \(2\): \(2^2\). With these two perfect squares, we identify that they are separated by a minus sign, indicating a difference of squares.

Using the difference of squares formula, we can then rewrite \(9z^2 - 4\) as \(3z + 2)(3z - 2)\), which demonstrates the application of this fundamental concept in algebra.

Factoring Expressions

Factoring expressions is a critical skill in algebra that simplifies polynomial expressions by expressing them as a product of their factors. Factoring makes it easier to solve equations, simplify fractions, and perform other algebraic operations. There are various methods for factoring expressions, but the goal is always to break down a complex expression into simpler ones that, when multiplied together, give back the original expression.

In our example, the initial expression \(18z^2 - 8\) is simplified in steps. By observing the common factor and the structure of the expression, the problem-solving approach becomes systematic and efficient. After extracting the common factor and identifying the difference of squares, it's clear that the expression can be factored further. Detailed step-by-step solutions, like the one provided for this exercise, are excellent tools for students to learn the methodical approach to factoring expressions.

Common Factor

Finding the common factor in an expression is a foundational aspect of factoring. It involves looking for a number or algebraic term that divides evenly into all terms of the expression. By factoring out this common factor, the expression becomes simpler and more manageable.

The problem we are examining begins with identifying 2 as the greatest common factor of \(18z^2\) and \(8\). This step is crucial because it allows for a reduction of the expression to its simplest form before addressing more complex patterns like the difference of squares. By extracting the common factor of 2, we obtain the intermediate expression \(2(9z^2 - 4)\), which is more approachable for further factoring. This strategy emphasizes the importance of seeking out and removing common factors as the first step in the factoring process.