Polynomial multiplication is a foundational topic in Algebra 1 and prerequisite of Algebra 2. It’s not only essential for simplifying expressions and solving equations, but also critical for students aiming to perform well in math competitions like the AMC 10.

In this guide, you’ll review must-know polynomial multiplication identities such as the difference of squares and perfect cubes. We’ll walk through examples from school math using the distributive property, and then explore how these same concepts appear in AMC-level problems that require deeper reasoning and strategic thinking.

What Is a Polynomial?

A polynomial is an algebraic expression that includes variables, constants, and exponents—all combined using addition, subtraction, multiplication or division. The exponents must be whole numbers.

You’ll find polynomials across most chapters of Algebra 1 and 2, especially in:

• Adding and subtracting polynomials
• Polynomial multiplication and factorization
• Long division and synthetic division

Knowing how to work with polynomials helps build the foundation for more advanced math, including functions, rational expressions, and even calculus.

Key Polynomial Multiplication Formulas

Understanding how to multiply polynomials is a key part of Algebra. These formulas not only make the process faster—they also help you recognize patterns and simplify complex expressions. Mastering them will give you a strong foundation for both school exams and math competitions.

Difference of Squares (In-school):

(a² – b²) = (a + b)(a – b)

Perfect Square Trinomials (In-school):

(a ± b)² = a² ± 2ab + b²

Square of a Sum of Three Terms (Math Competition):

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Symmetric Expansion (Math Competition):

a² + b² + c² + ab + bc + ca = ½[(a + b)² + (b + c)² + (c + a)²]

Sum of Cubes (Math Competition):

a³ + b³ = (a + b)(a² – ab + b²)

Difference of Cubes (Math Competition):

a³ – b³ = (a – b)(a² + ab + b²)

Perfect Cubes (Math Competition):

Knowing formulas is just the start—real math skill comes from knowing how to use them. In school, problems are direct. In contests like the AMC 10, they require deeper thinking and flexible problem-solving. Let’s look at how the same polynomial identities are used in both settings.

School-Level Polynomial Examples

Polynomial Example 1

Problem: Multiply −3x(6x + 1)

Solution:

In this case, multiply the monomial −3x by the binomial 6x + 1. Apply the distributive property.

Final Answer:

−18x² – 3x

Polynomial Example 2

Problem: Multiply -x²(3x² − 2x + 8)

Solution:

In this case, multiply the monomial −x² by the binomial 3x² − 2x + 8. Then apply the distributive property and simplify.

Final Answer:

-3x⁴ + 2x³ – 8x²

AMC 10-Style Application of Polynomial

Unlike school problems that give you the formula, AMC problems require you to recognize and apply it yourself. The two examples below are from the 2024 AMC 10A, Problems 11 and 15.

Polynomial Example 3

Problem: Let M be the greatest integer such that both M + 1213 and M + 3773 are perfect squares. What is the units digit of M?

Solution:

Step 1: Eliminate M and construct difference of squares pattern

Step 2: Solve a and b by casework

Step 3: Traditional Approach – Calculate M

But in a math competition, every second counts.

Here’s a faster method using modular arithmetic to find the units digit:

Step 3: Find the Units Digit

Answer:

Units digit of M = 8

Polynomial Example 4

Problem: How many ordered pairs of integers (m, n) satisfy: √(n² − 49) = m ?

Solution:

Factor pairs of 49:

Answer:

There are 4 integer solutions

Click here to download the complete solutions of 2024 AMC 10.

What’s the Difference of In-School and AMC 10 Polynomial Problems?

By comparing school-based and competition-style problems, here’s what we can observe:

The challenge in math competitions isn’t the formulas themselves, but how to apply them.

• School problems tend to be straightforward:
  • The question usually tells you exactly which formula or method to use.
  • You’re mainly practicing the skill directly.
• Competition problems require more logical and critical thinking:
  • You have to carefully read and interpret the problem.
  • Often, you need to figure out what formulas or strategies to apply based on context.
• Polynomial multiplication, while a major topic in Algebra 2,
  • Is often just a minor component of a competition problem.
  • You might use it as a step, but the main challenge could involve some other math concepts.
• Number theory often plays a big role in contest problems:
  • Reasoning about number parity
  • Figuring out maximum or minimum values
  • Understanding how numbers relate to each other

These number theory concepts aren’t usually covered in regular school math, so students aiming for competitions will need extra learning and practice beyond the classroom.

Ready to Practice?

🎯 Download your free Algebra worksheets with 10 polynomial practice questions.

✅ Score 100%? Take the Think Academy Algebra 2 Assessment for personalized 1-on-1 feedback.

💡 Interested in math competitions? Try our AMC 10 Evaluation Test to see where you stand.

The 2025 AMC 10 dates have just been released, and registration will open soon. Visit our AMC 10 program page to stay updated and register!

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