How to Use Area Models to Factor Variable Expressions?

An area model connects geometry and algebra by representing multiplication as the area of a rectangle. When you reverse this thinking, you can use an area model to factor variable expressions — that is, to write a sum or difference as a product. This approach makes factoring visual and intuitive, which is especially helpful on the GED Math test.

What Is an Area Model?

In an area model, a rectangle’s area equals its length times its width: \(\color{blue}{\text{ Area } = \text{ length } \times \text{ width }}\). If the area of a rectangle is expressed as a polynomial, the length and width are the factors of that polynomial.

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For example, if the area is \(\color{blue}{6x + 9}\), you can arrange tiles to form a rectangle whose dimensions reveal the factors. Here, a rectangle with width \(\color{blue}{3}\) and length \(\color{blue}{2x + 3}\) has area \(\color{blue}{3(2x + 3) = 6x + 9}\).

How to Use an Area Model to Factor

Step 1: Find the Greatest Common Factor (GCF)

The GCF of all terms becomes one dimension (side) of the rectangle. The other dimension is the remaining expression after dividing each term by the GCF.

Step 2: Set up the rectangle

Draw a rectangle. Place the GCF along one side. Divide the total area (the original expression) by the GCF to find the other side.

Step 3: Write the factored form

\(\color{blue}{\text{ Original expression } = \text{ GCF } \times (\text{ remaining expression })}\)

Inline Example: Factor \(\color{blue}{8x + 12}\)

• GCF of 8x and 12 is \(\color{blue}{4}\).
• Divide: \(\color{blue}{8x \div 4 = 2x}\) and \(\color{blue}{12 \div 4 = 3}\).
• Rectangle: \(\color{blue}{\text{ width } = 4}\), length = \(\color{blue}{(2x + 3)}\), area = \(\color{blue}{4(2x + 3)}\).
• Factored form: \(\color{blue}{8x + 12 = 4(2x + 3)}\).

Step-by-Step Summary

1. Identify all terms in the expression.
2. Find the GCF of the coefficients and any common variable factors.
3. Divide every term by the GCF to get the terms of the other factor.
4. Write the factored form: \(\color{blue}{\text{ GCF }(\text{ remaining terms })}\).
5. Verify by distributing the GCF back and confirming you get the original expression.

Watch: Factoring with an Area Model (Khan Academy)

Sal Khan uses the area model to factor expressions with a common factor:

Worked Examples

Example 1: Factor \(\color{blue}{6x + 9}\) using an area model.

\(\color{blue}{\text{ GCF }(6, 9) = 3}\). Divide: \(\color{blue}{6x \div 3 = 2x}\), \(\color{blue}{9 \div 3 = 3}\).
Factored: \(\color{blue}{3(2x + 3)}\). Check: \(\color{blue}{3 \times 2x + 3 \times 3 = 6x + 9}\) ✓

Example 2: Factor \(\color{blue}{10x – 15}\).

\(\color{blue}{\text{ GCF }(10, 15) = 5}\). Divide: \(\color{blue}{10x \div 5 = 2x}\), \(\color{blue}{15 \div 5 = 3}\).
Factored: \(\color{blue}{5(2x – 3)}\). Check: \(\color{blue}{5 \times 2x – 5 \times 3 = 10x – 15}\) ✓

Example 3: Factor \(\color{blue}{6x^{2} + 9x}\).

\(\color{blue}{\text{ GCF }(6x<\text{ sup }>2, 9x) = 3x}\). Divide: \(\color{blue}{6x^{2} \div 3x = 2x}\), \(\color{blue}{9x \div 3x = 3}\).
Factored: \(\color{blue}{3x(2x + 3)}\). Check: \(\color{blue}{3x \times 2x + 3x \times 3 = 6x^{2} + 9x}\) ✓

Example 4: Factor \(\color{blue}{12x – 8}\).

\(\color{blue}{\text{ GCF }(12, 8) = 4}\). Divide: \(\color{blue}{12x \div 4 = 3x}\), \(\color{blue}{8 \div 4 = 2}\).
Factored: \(\color{blue}{4(3x – 2)}\). Check: \(\color{blue}{4 \times 3x – 4 \times 2 = 12x – 8}\) ✓

More Practice: Factoring Using the Distributive Property (Khan Academy)

This Khan Academy lesson reinforces factoring as the reverse of the distributive property:

Exercises

Factor each expression using the area model (find the GCF and write factored form).

1. \(\color{blue}{4x + 8}\)
2. \(\color{blue}{9y – 12}\)
3. \(\color{blue}{15x + 25}\)
4. \(\color{blue}{8x^{2} + 6x}\)
5. \(\color{blue}{14n – 21}\)
6. \(\color{blue}{20a + 30b}\)

Answers

1. \(\color{blue}{4(x + 2)}\)
2. \(\color{blue}{3(3y – 4)}\)
3. \(\color{blue}{5(3x + 5)}\)
4. \(\color{blue}{2x(4x + 3)}\)
5. \(\color{blue}{7(2n – 3)}\)
6. \(\color{blue}{10(2a + 3b)}\)

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Frequently Asked Questions

What is the relationship between an area model and factoring?

In an area model, a rectangle’s area is the product of its dimensions. Factoring finds those dimensions. When the area is a polynomial expression, the factors are the side lengths of the rectangle.

How do I find the GCF for factoring?

List the factors of each term’s coefficient and identify the largest one they share. Also look for any common variable factors. The GCF is the product of the common numerical and variable factors at their lowest powers.

Can I use the area model for three or more terms?

Yes. For three terms, you draw a rectangle divided into three sections and find the GCF across all three terms. The process is the same: GCF forms one dimension and the remaining terms form the other.

Related Topics

• Using Distributive Property to Factor Variable Expressions
• Use Properties to Write Equivalent Expressions
• Identify Equivalent Expressions
• Use Algebra Tiles to Identify Equivalent Expressions
• Identifying Expressions and Equations