Problem 41 - Algebra A Combined Function

Question

Factor each completely. $$ 18 x^{2} y-2 y $$

Step-by-Step Solution

Answer

The expression \(18x^2y - 2y\) factors completely to \(2y(3x + 1)(3x - 1)\).

Step 1: Identify the Common Factor

Look for the greatest common factor (GCF) in the expression \(18x^2y - 2y\). Both terms contain the factor \(2y\).

Step 2: Factor Out the GCF

Factor \(2y\) out of the expression. This means you divide each term by \(2y\): \[ 18x^2y \div 2y = 9x^2 \] \[ -2y \div 2y = -1 \] This gives you: \[ 2y(9x^2 - 1) \]

Step 3: Recognize the Difference of Squares

Observe that the expression inside the parentheses \( 9x^2 - 1 \) is a difference of squares, as it can be written as \((3x)^2 - 1^2\).

Step 4: Apply Difference of Squares Formula

Use the difference of squares formula \( a^2 - b^2 = (a+b)(a-b) \), where \( a = 3x \) and \( b = 1 \): \[ (3x + 1)(3x - 1) \] Thus, the full factored expression becomes: \[ 2y(3x + 1)(3x - 1) \]

Step 5: Confirm Your Solution

Multiply the factors back together to check your work: First, multiply \((3x + 1)(3x - 1)\): \[ (3x + 1)(3x - 1) = 9x^2 - 1 \] Then, multiply by \(2y\): \[ 2y(9x^2 - 1) = 18x^2y - 2y \] This confirms that the factorization is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more terms in an expression. For the problem given, we have the expression \(18x^2y - 2y\). The GCF is crucial because factoring it out simplifies the expression significantly. To find the GCF:

• List all factors of each term. For \(18x^2y\), the factors are \(2 \times 3^2 \times x^2 \times y\).
• For \(2y\), the factors are \(2 \times y\).
• Identify the common factors in both terms, which here are \(2y\).

Once you determine that \(2y\) is the GCF, you can factor it out from each term:\[18x^2y \div 2y = 9x^2, \quad -2y \div 2y = -1\]Essentially, it simplifies the expression to \(2y(9x^2 - 1)\). Pulling out the GCF is often the first and most important step in simplifying algebra expressions, making further factorization steps easier.

Difference of Squares

The concept of difference of squares involves recognizing and applying a particular identity: \(a^2 - b^2 = (a+b)(a-b)\). In this exercise, after factoring out the GCF, we are left with \(9x^2 - 1\). This expression is a classic example of a difference of squares because:

• \(9x^2\) is \((3x)^2\)
• \(1\) is \(1^2\)

The identity can now be applied by setting \(a = 3x\) and \(b = 1\), yielding:\[(3x + 1)(3x - 1)\]This treatment of expressions into a difference of squares format is a powerful algebra tool. It reduces quadratic expressions to the product of two binomials, simplifying solving equations or further algebraic manipulation.

Algebra Problems

Algebra problems, such as the one discussed, often involve multiple steps requiring different algebraic techniques. They are about understanding patterns and applying appropriate methods. Let's break down the steps with focus on applying strategies efficiently:

• **Identify Simple Patterns:** Start by finding the greatest common factor and factored out the GCF.
  
• **Recognize Special Forms:** Notice patterns like difference of squares, which can be transformed and solved using specific formulas.
• **Verify Solutions:** Always multiply factors back to the original expression to check the correctness. These problems strengthen skills through verifying reasoning and results.

Solving such algebra problems builds foundational skills in mathematical reasoning and helps in tackling more complex problems by step-by-step logical application of algebraic principles.