How to Use Area Models to Factor Variable Expressions?

How to use area models to factor variable expressions, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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. \(\color{blue}{10(2a + 3b)}\)

Frequently Asked Questions

What is factoring in math?

Factoring means rewriting an expression as a product of simpler pieces, called factors. It is the reverse of multiplying. Since 3(2x + 3) = 6x + 9, factoring 6x + 9 gives 3(2x + 3). In an area model, the factors are the length and width of a rectangle, and the expression you start with is its area.

How do you factor using an area model?

Treat the expression as the area of a rectangle and find its side lengths. Start with the greatest common factor of all the terms and place it on one side. Divide each term by the GCF to get the other side. For 10x + 15, the GCF is 5, the other side is 2x + 3, so 10x + 15 = 5(2x + 3).

What are the steps to factoring with an area model?

Step 1: find the GCF of every term. Step 2: draw a rectangle and write the GCF along one side. Step 3: divide each term of the expression by the GCF and write the results along the other side. Step 4: write the answer as GCF × (other side). Check by multiplying back: 4(3x − 2) = 12x − 8.

How do you use an area model for factoring polynomials?

Place each term of the polynomial in the rectangle, then find the side that is common to all of them. For 4x² + 8x, both terms share 4x, so one side is 4x. Dividing gives 4x² ÷ 4x = x and 8x ÷ 4x = 2, so the other side is x + 2. The factored form is 4x(x + 2).

How do you factor trinomials with an area model?

Use a 2 by 2 grid. Put the x² term in the top-left box and the constant in the bottom-right box. Then split the middle term into two parts that fit both. For x² + 5x + 6, the boxes hold x², 2x, 3x and 6, so the sides are x + 2 and x + 3, giving (x + 2)(x + 3).

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