Multiply Polynomials Worksheet: Free Printable PDF + Lesson

Most students can multiply two binomials with a little practice. The trouble starts when the polynomials get bigger — a trinomial times a binomial, or squaring a binomial — and the shortcut they memorised stops working. I have seen this exact gap trip up students right before their algebra exam, and it is entirely preventable.

Multiplying polynomials is the process of applying the distributive property to find the product of two or more polynomial expressions. Every term in the first polynomial multiplies every term in the second. The result is simplified by combining like terms and written in standard form (highest degree first).

• Understand why the distributive property is the only rule you truly need.
• Apply the FOIL method correctly — and know exactly when it breaks down.
• Spot and fix the four most common multiplication errors.
• Complete 10 graded practice problems and check your work with the answer key.
• Download the free printable PDF worksheet to use offline or in class.

What Are Polynomials, and Why Does Multiplying Them Matter?

A polynomial is an expression made of one or more terms, where each term is a number, a variable, or a product of numbers and variables with whole-number exponents. Examples: 3x, x^2 + 5, 2x^3 – 4x + 1. Multiplying polynomials is the skill of finding the product of two such expressions.

This matters far beyond the algebra classroom. Polynomial multiplication is the direct prerequisite for factoring (you cannot factor what you cannot first expand), solving quadratic equations, working with polynomial division, and understanding calculus concepts like the product rule for derivatives. In my experience teaching algebra, students who struggle to factor quadratics almost always have a gap in polynomial multiplication — not factoring itself.

How to Multiply Polynomials: The Step-by-Step Method That Always Works

The distributive property is the only rule you need for all polynomial multiplication. Every other technique — FOIL, the box method, vertical multiplication — is just a way of organising the same underlying process.

1. Write both polynomials clearly in parentheses.
   Place them side by side: (first polynomial)(second polynomial). This visual separation prevents sign errors.
2. Pick one term from the first polynomial and distribute it across every term in the second.
   Multiply that single term by each term of the second polynomial in order, left to right.
3. Repeat for every remaining term in the first polynomial.
   Each term in the first polynomial gets its own row of partial products. For a binomial times a binomial, you get two rows of two products each (4 total). For a trinomial times a binomial, three rows of two (6 total).
4. Multiply coefficients and apply the exponent rule.
   When multiplying variable terms: multiply the numbers out front, then add the exponents of matching variables. Example: (3x^2)(4x^3) = 12x^5.
5. Collect and combine like terms.
   Group terms that share the same variable and exponent. Add or subtract their coefficients. A term with x^2 and a term with x are NOT like terms — do not combine them.
6. Write the answer in standard form.
   Arrange terms from highest degree to lowest. The constant (no variable) goes last.

Three Fully Worked Examples — From Basic to Advanced

Each example below uses the same five-step method. Work through them in order — the difficulty increases deliberately.

  FOIL Diagram: (2x + 3)(x + 5)
  ┌─────────────────────────────────────┐
  │  F:  2x * x   =  2x²               │
  │  O:  2x * 5   =  10x  ─┐           │
  │  I:   3 * x   =   3x  ─┴─ 13x      │
  │  L:   3 * 5   =   15               │
  │                                     │
  │  Result:  2x² + 13x + 15           │
  └─────────────────────────────────────┘
  

Common Mistakes When Multiplying Polynomials — and the Right Fix

These four errors account for the majority of wrong answers I see on polynomial multiplication problems. Recognise them now so you do not make them on a test.

On-Page Practice Worksheet: 10 Problems, Easy to Hard

How to use this worksheet: work through each problem on paper, show all your partial products, then check your answers using the collapsible key below. If you get a problem wrong, go back to the worked examples and identify which step you missed. Print the PDF version using the download button for offline or classroom use.

1. (x + 2)(x + 3)
2. (x – 4)(x + 1)
3. (2x + 3)(x + 5)
4. (3x – 2)(x – 4)
5. (x + 6)^2
6. (2x – 5)^2
7. 3x(x^2 – 4x + 7)
8. (x + 2)(x^2 – 3x + 4)
9. (2x – 1)(3x^2 + x – 5)
10. (x^2 + 3x – 2)(x^2 – x + 1)