Difference of Squares: How to Factor Polynomials Step-by-Step

Your child stares at (x² – 49) and asks, “Why can’t I just multiply?” It’s a common scene in many households. Factoring polynomials can seem tricky at first, but there’s a neat trick called difference of squares that makes it simple. Once kids understand it, many homework problems feel like solving a puzzle rather than a chore. Let’s break it down together and make math fun at home.

What Is the Difference of Squares?

The difference of squares occurs when you subtract one perfect square from another.

• Formula: a² – b² = (a – b)(a + b)

Think of it as a large square cake. If you remove a smaller square from one corner, you can rearrange the remaining pieces into two rectangles. This visual helps kids see why the formula works, instead of just memorizing it.

Examples:

1. x² – 16 = (x – 4)(x + 4)
2. 49 – y² = (7 – y)(7 + y)
3. 9a² – 25b² = (3a – 5b)(3a + 5b)

Each example shows how subtraction of squares turns into a simple multiplication of two factors.

How to Do Difference of Squares Step-by-Step

Step 1: Identify Two Perfect Squares

Check if both terms are perfect squares, either numbers or variables.

• Example 1: (x² – 25) ✅ both perfect squares
• Example 2: (x² – 10) ❌ not a perfect square

Tip: Remember, (any number)² and (any variable)² are perfect squares.

Step 2: Write in a standard form

Once you spot the squares, apply the formula a² – b² = (a – b)(a + b)

• Example 1 (Numbers): 36 – 9 = (6 – 3)(6 + 3) = 3 x 9 = 27
• Example 2 (Variables): x² – 49 = (x – 7)(x + 7)
• Example 3 (Coefficients & Variables): 9x² – 4y² = (3x – 2y)(3x + 2y)

Tip: Always simplify coefficients by taking square roots first.

Step 3: Verify Your Answer

Multiply your factors to check if you return to the original expression. This step builds confidence and avoids common mistakes.

Common Mistakes & Myth-Busting

It’s easy for kids to confuse similar formulas. Here’s a quick guide:

Tip for parents: Walk your child through multiplication after factoring. Seeing the original expression come back helps solidify understanding.

Common Perfect Squares

Before factoring difference of squares, it helps to recognize common perfect squares. A perfect square is any number multiplied by itself. Here’s a handy list for kids:

Try It at Home with Your Kid

Make factoring interactive. Here are three hands-on activities:

1. Kitchen Squares:
   Use cookie sheets, cutting boards, or even tiles. Represent (a² – b²) with physical squares. Let your child cut a smaller square out of a larger one and rearrange into rectangles. Discuss how the area stays the same.
2. Paper Puzzle:
   Draw a large square on paper. Draw and cut a smaller square inside it. Move the remaining pieces to form two rectangles. Label each piece with algebraic expressions to visualize factoring.
   
3. Calculator Check:
   Factor expressions like (x² – 64) on paper, then use a calculator to multiply factors and see the original number. This reinforces both arithmetic and algebraic reasoning.

Extra Fun: Turn it into a game: “Who can spot the perfect squares fastest?” or “Recreate this square puzzle using colored post-it notes.”

More Original Practice Examples

• (x² – 121 = ?)
• (4y² – 49 = ?)
• (16a²b² – 25c² = ?)

Solutions:

1. (x – 11)(x + 11)
2. (2y – 7)(2y + 7)
3. (4ab – 5c)(4ab + 5c)

Encourage your child to verify each solution by multiplying the factors.

Learn More with WuKong Math

Understanding square numbers is just the beginning of a child’s exciting math journey. To help children build a strong foundation, WuKong Math offers a comprehensive online learning program designed for K–12 students.

Our experienced teachers make complex math ideas, like squares, fractions, and algebra, easy to grasp through engaging lessons, visual explanations, and real-world problem-solving. With WuKong Math, students don’t just memorize formulas; they learn why math works and how to apply it with confidence.

WuKong’s curriculum follows international standards and encourages logical thinking, creativity, and mathematical fluency. Whether your child is learning basic arithmetic or preparing for advanced math, WuKong Math provides the personalized support they need to succeed.

Conclusion

The difference of squares is a single trick that unlocks a wide range of algebra problems. Trying a few examples tonight with your child can turn frustration into excitement. Once kids see the pattern, factoring becomes a skill they can use confidently across math and science.

FAQ

James | WuKong Math Teather

Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions’ teaching, including Math Kangaroo, AMC… He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.