Problem 58 - Algebra 1: Concepts and Skills

Question

Completely factor the expression. $$ 36 x^{3}-9 x $$

Step-by-Step Solution

Answer

The completely factorized expression is \(9x(2x-1)(2x+1)\).

Step 1: Identify Common Factors

Find the greatest common factor (GCF) of the two terms. The GCF of \(36x^3\) and \(9x\) is \(9x\). So, factor out the GCF from the original equation. The equation becomes \(9x(4x^2 - 1)\).

Step 2: Apply the Difference of Square Method

Identify \(4x^2 - 1\) as difference of squares and apply the formula \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a=2x\) and \(b=1\). After applying, we have \(9x[(2x-1)(2x+1)]\).

Step 3: Write Down the Final Factorized Expression

Consolidate all these separate factors to write down the final completely factorized expression. The original expression \(36x^3 - 9x\) factorizes to \(9x(2x-1)(2x+1)\).

Key Concepts

Greatest Common Factor

Factoring algebraic expressions often starts with finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in an expression without leaving a remainder.
For example, in the expression \(36x^3 - 9x\), the GCF is crucial for simplifying the terms.

• First, list the factors of each term: \(36x^3\) has factors including \(1, 3, 9, 12, 18, 36, x, x^2, x^3\), and \(9x\) has factors \(1, 3, 9, x\).
• The common factors are \(1, 3, 9, x\), and the greatest among them is \(9x\).

Finding the GCF allows us to factor an expression more efficiently, reducing it to a simpler form. In this expression, factoring out \(9x\) results in \(9x(4x^2 - 1)\). This step is essential as it simplifies the expression and sets the stage for further factoring methods.

Difference of Squares

After factoring out the greatest common factor, it's important to look for patterns like the difference of squares. The difference of squares is a particular algebraic pattern where a squared term is subtracted from another squared term, taking the form \(a^2 - b^2\).
This pattern can be rewritten as \((a-b)(a+b)\). For instance, in the expression \(4x^2 - 1\), we see:

• \(a = 2x\)
• \(b = 1\)

Both \(2x\) and \(1\) are squared in the terms \(4x^2\) and \(1\).
The expression then becomes \((2x-1)(2x+1)\) through the difference of squares method. Identifying and using the difference of squares technique helps in simplifying these problems, making it easier to see the factorized form of the original expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis for much of algebra. Understanding how to manipulate these expressions through factoring is key to simplifying complex problems.
In the context of our example, \(36x^3 - 9x\) is an algebraic expression. Here's how it can be factored:

• First, notice the greatest common factor, which is \(9x\).
• Factor it out to get a simpler expression, \(9x(4x^2 - 1)\).
• Recognize the expression \(4x^2 - 1\) as a difference of squares.
• Factor it further into \((2x-1)(2x+1)\).

These steps showcase common techniques used for handling algebraic expressions.
By breaking down the expression into multiple layers of simpler factors, you can work towards the complete factorized form efficiently. Understanding these principles can greatly aid in mastering algebra and solving equations effectively.