| Detail | Info |
|---|---|
| Skill | Multiplying polynomials (binomials, trinomials, higher-degree) |
| Grade Level | Algebra 1 (Gr. 8-9) & Algebra 2 (Gr. 10-11) |
| Core Method | Distributive property; FOIL shortcut for binomials |
| Problems in Worksheet | 10 (easy to hard) |
| Answer Key | Included (on-page + PDF) |
| Download | Free printable PDF (see buttons below) |
| Pre-requisite | Understanding of exponent rules and combining like terms |
Multiplying Polynomials Worksheet: Free PDF, Worked Examples & Answer Key
Table of Contents
- Introduction
- Quick Answer
- Download the Free PDF Worksheet
- TL;DR Summary
- What Are Polynomials?
- Why Multiplying Polynomials Matters
- Step-by-Step: How to Multiply Polynomials
- Fully Worked Examples
- FOIL vs. Box Method: Which Should You Use?
- Common Mistakes (and How to Avoid Them)
- Unique Insight
- On-Page Practice Worksheet
- Quick Quiz
- FAQ
- Key Takeaways
- Related Articles
- About the Author
- Sources & References
If you have searched for a multiplying polynomials worksheet, you probably want two things: a clear explanation of how the skill works, and a set of graded problems you can print and practice right now. This page gives you both — plus a free downloadable PDF with a full answer key.
Multiplying polynomials is the process of applying the distributive property to find the product of two or more polynomial expressions. For binomials, the FOIL method (First, Outer, Inner, Last) organizes the four required multiplications. For larger polynomials, you extend the same distributive logic to every pair of terms, then collect like terms to simplify.
After working through this page you will be able to:
- Explain the distributive property as it applies to polynomial multiplication.
- Use FOIL correctly for any two binomials.
- Multiply a binomial by a trinomial without missing a term.
- Recognize and apply the perfect-square and difference-of-squares shortcuts.
- Avoid the four most common errors students make on tests.
Free Printable PDF: 10 graded problems on multiplying polynomials, with a full answer key on the back. No sign-up required.
⚡ TL;DR — Quick Summary
- Polynomial multiplication = distribute every term, then combine like terms.
- FOIL works only for binomial × binomial (four products).
- Binomial × trinomial gives six products; use full distribution.
- Perfect square pattern: (x+a)² = x² + 2ax + a².
- Difference of squares: (x+a)(x-a) = x² – a².
- The box method is safer than FOIL for polynomials with 3+ terms.
What Are Polynomials, and What Does It Mean to Multiply Them?
A polynomial is an algebraic expression made up 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: 3x² - 5x + 2 is a trinomial (three terms); x + 7 is a binomial (two terms).
Multiplying two polynomials means finding a single polynomial that equals their product. The degree of the result equals the sum of the degrees of the two factors. So a degree-2 polynomial times a degree-3 polynomial produces a degree-5 result.
Why Does Multiplying Polynomials Matter?
Polynomial multiplication is a gateway skill. You cannot factor quadratics, solve polynomial equations, or work with rational expressions without it. It also appears directly on the SAT, ACT, and most state algebra assessments.
In my experience teaching algebra, students who struggle to factor quadratics almost always trace the problem back to shaky polynomial multiplication. Factoring is just multiplication in reverse — if you cannot multiply fluently, factoring feels like guesswork.
Beyond tests, polynomial multiplication models real-world quantities: area calculations, physics equations for motion, and engineering formulas all involve products of polynomial expressions.
In my experience, the single biggest reason students lose points on polynomial problems is not a lack of understanding — it is a lack of a systematic process. They try to do FOIL in their head and miss the outer or inner term. Writing out every partial product on a separate line, even when it feels slow, eliminates almost all of those errors. Speed comes with practice; accuracy has to come first.
How Do You Multiply Polynomials? Step-by-Step Guide
The process below works for any polynomial multiplication. FOIL is a special case of Step 2 for two binomials.
- Write both polynomials in standard form (descending powers of the variable). This prevents sign errors and makes combining like terms easier.
- Distribute every term in the first polynomial across every term in the second. If the first has m terms and the second has n terms, you will get exactly m × n partial products.
- Apply exponent rules when multiplying variable terms: x² × x³ = x⁵ (add the exponents).
- Watch the signs. Negative × negative = positive. Write the sign explicitly on every partial product before moving on.
- Collect like terms. Group terms with the same variable and exponent, then add their coefficients.
- Verify the degree. The degree of your answer should equal the sum of the degrees of the two factors.
( 2x + 3 ) ( x - 5 )
| | | |
|___________| | | First: 2x * x = 2x²
|________________| | Outer: 2x * (-5) = -10x
|_________| Inner: 3 * x = 3x
Last: 3 * (-5) = -15
Sum: 2x² -10x +3x -15
↓
Result: 2x² - 7x - 15
Fully Worked Examples: Multiplying Polynomials
Work through each example carefully. The method shown here matches the problems in the worksheet below.
Example 1 — Binomial × Binomial (FOIL): (x + 4)(x – 3)
Step 1 — First: x * x = x²
Step 2 — Outer: x * (-3) = -3x
Step 3 — Inner: 4 * x = 4x
Step 4 — Last: 4 * (-3) = -12
Sum: x² - 3x + 4x - 12
Combine like terms: -3x + 4x = x
Answer: x² + x - 12
Example 2 — Perfect Square Binomial: (2x – 3)²
Rewrite: (2x - 3)(2x - 3)
First: 2x * 2x = 4x²
Outer: 2x * (-3) = -6x
Inner: (-3) * 2x = -6x
Last: (-3) * (-3) = 9
Sum: 4x² - 6x - 6x + 9
Combine: -6x - 6x = -12x
Answer: 4x² - 12x + 9
Shortcut check: (a - b)² = a² - 2ab + b²
a=2x, b=3 → 4x² - 12x + 9 ✓
Example 3 — Binomial × Trinomial: (x + 4)(x² – 2x + 3)
Distribute x across the trinomial:
x * x² = x³
x * (-2x) = -2x²
x * 3 = 3x
Distribute 4 across the trinomial:
4 * x² = 4x²
4 * (-2x) = -8x
4 * 3 = 12
All six products:
x³ -2x² +3x +4x² -8x +12
Combine like terms:
x³ + (-2x² + 4x²) + (3x - 8x) + 12
x³ + 2x² + (-5x) + 12
Answer: x³ + 2x² - 5x + 12
FOIL vs. Box Method: Which Should You Use?
Both methods apply the distributive property. The choice depends on the size of the polynomials and your personal preference.
| Feature | FOIL Method | Box (Grid) Method |
|---|---|---|
| Works for binomial × binomial | Yes (designed for it) | Yes |
| Works for binomial × trinomial | No (misses terms) | Yes (2 × 3 grid) |
| Works for trinomial × trinomial | No | Yes (3 × 3 grid) |
| Visual organization | Linear — easy to lose a term | Grid — hard to miss a cell |
| Speed (once mastered) | Faster for binomials | Slightly slower but safer |
| Best for | Quick binomial products | Any polynomial size |
I teach FOIL first because it is fast and the mnemonic sticks. But I always follow up with the box method for anything beyond binomials. In my experience, students who learn only FOIL get into trouble the moment they see a trinomial factor — they try to force FOIL and drop terms. Learning both gives you a reliable fallback for every case you will encounter on a test.
Common Mistakes When Multiplying Polynomials (and How to Fix Them)
These are the four errors I see most often in student work. Each one has a simple fix.
Wrong: (x + 3)(x + 5) = x² + 15
Right: x² + 5x + 3x + 15 = x² + 8x + 15
Fix: Write all four FOIL products on separate lines before adding.
Wrong: (x – 4)(x + 2) → Last term: (-4)(2) = 8
Right: (-4)(+2) = -8, so answer is x² – 2x – 8
Fix: Circle every negative sign before you start multiplying.
Wrong: 3x² × 2x = 5x³ (added coefficients)
Right: 3x² × 2x = 6x³ (multiply coefficients, add exponents)
Fix: Treat the number part and the variable part separately.
Wrong: x² + 5x + 3x + 15 left as four separate terms
Right: x² + 8x + 15
Fix: Always scan for matching variable-exponent pairs before writing the final answer.
Most worksheet sites teach FOIL as if it is a separate rule from the distributive property. That framing backfires: when students hit a binomial × trinomial problem, they think FOIL “doesn’t apply” and freeze. The truth is there is only one rule — distribute every term across every term. FOIL is just a memory aid for counting the four products in the specific case of two binomials.
A more useful mental model: think of polynomial multiplication as an area problem. If you have a rectangle with sides (x + 3) and (x + 5), the area is the sum of four smaller rectangles: x², 5x, 3x, and 15. This geometric picture makes it obvious why you need all four products — you are accounting for every sub-region of the rectangle. Students who visualize it this way almost never miss a term.
On-Page Practice Worksheet: Multiplying Polynomials (10 Problems)
These 10 problems progress from straightforward binomial products to higher-degree polynomial multiplication. Work through them in order. Use the FOIL method for problems 1-6 and the full distributive property for problems 7-10.
How to use this worksheet: Print the PDF (button below) or work directly on paper. Attempt every problem before checking the answer key. If you get a problem wrong, go back to the worked example that matches its type and identify where your process broke down.
- (x + 3)(x + 5)
- (x – 4)(x + 2)
- (2x + 1)(x + 6)
- (3x – 2)(x – 5)
- (x + 7)²
- (2x – 3)²
- (x + 4)(x² – 2x + 3)
- (2x – 1)(3x² + x – 4)
- (x² + 3x – 2)(x² – x + 1)
- (3x² – 2x + 4)(2x² + x – 3)
Show Answer Key
- x² + 8x + 15
- x² – 2x – 8
- 2x² + 13x + 6
- 3x² – 17x + 10
- x² + 14x + 49
- 4x² – 12x + 9
- x³ + 2x² – 5x + 12
- 6x³ – x² – 9x + 4
- x⁴ + 2x³ – 4x² + 5x – 2
- 6x⁴ – x³ + 5x² – 14x – 12
Problems 9 and 10 are challenging — they are designed to stretch Algebra 2 students. If you are in Algebra 1, focus on problems 1-7 first.
Download the full worksheet as a printable PDF — includes all 10 problems plus the answer key on a separate page, formatted for standard letter paper.
Reveal-on-Click Practice Problems
Try each problem, then click to see the full worked solution.
Practice A: Multiply (x + 6)(x – 6)
This is a difference of squares: (a + b)(a – b) = a² – b².
a = x, b = 6
Answer: x² - 36
Verify with FOIL:
First: x * x = x²
Outer: x * (-6) = -6x
Inner: 6 * x = 6x
Last: 6 * (-6) = -36
Sum: x² - 6x + 6x - 36 = x² - 36 ✓
Practice B: Multiply (3x + 2)(2x² – x + 4)
Binomial × trinomial: 2 × 3 = 6 partial products.
Distribute 3x:
3x * 2x² = 6x³
3x * (-x) = -3x²
3x * 4 = 12x
Distribute 2:
2 * 2x² = 4x²
2 * (-x) = -2x
2 * 4 = 8
All products: 6x³ - 3x² + 12x + 4x² - 2x + 8
Combine like terms:
x³: 6x³
x²: -3x² + 4x² = x²
x: 12x - 2x = 10x
constant: 8
Answer: 6x³ + x² + 10x + 8
Practice C: Expand (x² + 2)(x² – 5)
Treat x² as a single unit and apply FOIL.
Let u = x²:
(u + 2)(u - 5)
= u² - 5u + 2u - 10
= u² - 3u - 10
Substitute back u = x²:
= (x²)² - 3(x²) - 10
= x⁴ - 3x² - 10
Answer: x⁴ - 3x² - 10
Quick Quiz: Test Your Understanding
Q1. What is (x + 5)(x – 2)?
Show Answer
Sources & References
Written and fact-checked by Dr Irfan Mansuri. External links open in a new tab and are provided for further reading and verification.