Polynomial Operations Worksheet: Add, Subtract, and Multiply (With Answer Key)

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Five free printables that take Algebra I and II students from naming the parts of a polynomial through adding, subtracting, multiplying four different ways, and recognizing special products. Fifty questions, a teacher answer key with every prompt restated, and the one checking habit that catches most of the errors before the paper gets turned in.

Written by Berke Sahbazoglu · BA Biochemistry, Washington University in St. Louis · 10+ years tutoring Algebra I & II · Burke Tutoring in Fremont · Free to print · Prepared from previous class notes and assigned homework, and checked for accuracy before publishing.

Published August 7, 2026 · Updated August 7, 2026

The parts of a polynomial

A polynomial is a sum of terms, and each term is a number multiplied by a variable raised to a whole-number power. In 5x⁴ − 3x² + 7x − 9 there are four terms. The coefficients are 5, −3, and 7. The constant term is −9. The degree is 4, because that is the largest exponent once the expression is simplified, and the leading coefficient is the 5 that sits in front of it.

Two vocabulary habits save students trouble later. First, the sign in front of a term belongs to that term: the second term above is −3x², not 3x². Second, degree is decided after simplifying, so 4x³ − 4x³ + x has degree 1, not 3.

Terms	Name by number of terms	Example
1	Monomial	7x3
2	Binomial	−4x3 + 2x
3	Trinomial	x2 − 2x + 3

Standard form and like terms

Standard form runs from the highest power down to the lowest. Writing 7 − 2x³ + 5x − x⁴ as −x⁴ − 2x³ + 5x + 7 is not busywork: the leading coefficient, the degree, and the end behavior of the graph all read straight off the front of the expression once it is ordered.

Like terms need the same variables raised to the same powers. That is why 4x² and −7x cannot be combined, and why 3x² + 5x is already finished. The most common error on this topic is a student writing 3x² + 5x = 8x³, adding both the coefficients and the exponents across terms that were never alike in the first place.

Adding and subtracting polynomials

Addition is just combining like terms. Subtraction is where the marks come off, and almost always in the same place: students flip the sign of the first term inside the parentheses and forget the rest.

When the two polynomials have different degrees, zero placeholders keep the columns honest. Adding (3x⁵ − 2x² + 4) and (x⁴ + 6x − 9) is easier if the missing powers are written in as 0x⁴, 0x³, and so on. The answer, 3x⁵ + x⁴ − 2x² + 6x − 5, then falls out column by column.

Four ways to multiply polynomials

FOIL is not a separate law of multiplication. It is a memory device for the four products created in a binomial-by-binomial distribution. The box and vertical layouts organize those same products differently, and both keep working when FOIL runs out of letters. Here is (x + 3)(x − 2) worked four ways, with the same answer each time.

Method	Best used for	What goes wrong
Distributive property	Any two polynomials, any size	Skipping a term once there are more than four products
FOIL	Two binomials, nothing else	Forcing FOIL onto a trinomial, which only has a name for four of the six products
Box method	Visual organization; larger products	Filling the grid correctly, then forgetting to combine like terms
Vertical multiplication	Long polynomials; familiar arithmetic layout	Misaligning powers or omitting zero placeholders

Special products worth recognizing

Three patterns show up often enough to be worth knowing on sight. None of them is a new rule. Each is ordinary distribution with the writing already done.

Pattern	Expanded	Example
Square of a sum	(a + b)2 = a2 + 2ab + b2	(x + 5)2 = x2 + 10x + 25
Square of a difference	(a − b)2 = a2 − 2ab + b2	(x − 7)2 = x2 − 14x + 49
Difference of squares	(a + b)(a − b) = a2 − b2	(x + 6)(x − 6) = x2 − 36

The error to watch for is (x − 4)² = x² + 16. Squaring a binomial always produces a middle term, because the two cross products are identical rather than opposite: −4x and −4x add to −8x. A difference of squares only appears when the cross products cancel, which happens when the two factors are conjugates.

The substitution check

Standards covered

Worksheet	Standard	What the standard asks for
Foundations	A-SSE.A.1a	Interpret the parts of an expression: terms, factors, and coefficients
Adding and subtracting	A-APR.A.1	Add, subtract, and multiply polynomials, and understand that they are closed under those operations
Multiplying polynomials	A-APR.A.1	The same closure standard, applied to products of larger polynomials
Special products	A-APR.C.4, A-SSE.A.2	Prove and use polynomial identities; use structure to rewrite an expression

Full wording for each standard is on the Common Core algebra standards page. If you want a task-based companion for the same standards, the Illustrative Mathematics task bank pairs well with the multiplication worksheet, and Desmos is a quick way to have students confirm that a factored form and an expanded form graph as the same curve.

Practice problems

Twenty problems drawn from across the five printables. The full set of fifty is in the downloads below.

Part A · Vocabulary and standard form

1. For 5x⁴ − 3x² + 7x − 9, list the terms, leading coefficient, constant term, and degree.
2. Classify −4x³ + 2x by number of terms and by degree.
3. Write in standard form: 7 − 2x³ + 5x − x⁴.
4. Combine like terms: 6x² − 3x + 8 + 5x − 2x² − 11.
5. Let P(x) = 2x³ − 5x + 1. Find P(−2).

Part B · Adding and subtracting

6. (3x² + 5x − 4) + (2x² − 7x + 9)
7. (5x² + 2x − 7) − (3x² − 6x + 4)
8. (2y³ − y² + 4y − 9) − (−3y³ + 5y² − 2y + 1)
9. A triangle has side lengths 2x + 3, x² − 1, and 3x² − x + 5. Write its perimeter in standard form.
10. A polynomial A(x) satisfies A(x) + (2x² + x − 7) = 5x² − 3x − 5. Find A(x).

Part C · Multiplying

11. Distribute: −2a³(5a² − 3a + 4)
12. Multiply: (2x − 3)(x + 5)
13. Use a 2 × 2 box for (3x + 2)(2x − 7). Show all four products.
14. Use vertical multiplication: (x − 5)(x² + 2x + 3)
15. Challenge: (2x² − 3x + 1)(x² + x − 4)
16. Why does FOIL not directly organize (x + 2)(x² − 3x + 4)? Name one method that does.

Part D · Special products

17. Expand: (2x + 3)²
18. Expand: (5a + 2)(5a − 2)
19. Factor as a special product: x² + 18x + 81
20. Jordan writes (x − 4)² = x² + 16. Which term is missing? Write the correct expansion.

Answer key

Part A · Vocabulary and standard form

1. Terms 5x⁴, −3x², 7x, −9; leading coefficient 5; constant −9; degree 4
2. Binomial; cubic (degree 3)
3. −x⁴ − 2x³ + 5x + 7
4. 4x² + 2x − 3
5. 2(−2)³ − 5(−2) + 1 = −16 + 10 + 1 = −5

Part B · Adding and subtracting

6. 5x² − 2x + 5
7. 2x² + 8x − 11
8. 5y³ − 6y² + 6y − 10
9. 4x² + x + 7
10. A(x) = (5x² − 3x − 5) − (2x² + x − 7) = 3x² − 4x + 2

Part C · Multiplying

11. −10a⁵ + 6a⁴ − 8a³
12. 2x² + 7x − 15
13. Cells 6x², −21x, 4x, −14; result 6x² − 17x − 14
14. x³ − 3x² − 7x − 15
15. 2x⁴ − x³ − 10x² + 13x − 4
16. FOIL names four products, which assumes two terms in each factor. Here there are 2 · 3 = 6 products. Distribution, a 2 × 3 box, or vertical multiplication all work.

Part D · Special products

17. 4x² + 12x + 9
18. 25a² − 4
19. (x + 9)²
20. Missing term −8x. Correct expansion x² − 8x + 16, because the two cross products are both −4x.

Download the printables

Twelve pages in total: four student worksheets, a one-page method guide for the binder, and a two-page teacher key that restates every prompt so it can be graded without the worksheets in hand. Free to print and copy for classroom use.

Choose which files to download

Foundations worksheet — 2 pages, 12 questions Adding and subtracting worksheet — 2 pages, 12 questions Four ways to multiply — 1-page method guide Multiplying polynomials worksheet — 3 pages, 14 questions Special products worksheet — 2 pages, 12 questions Teacher answer key — 2 pages, all 50 questions

Download selected Get the 12-page bundle instead

Questions teachers ask

Should I still teach FOIL?

Teach it as a name, not as a method. FOIL is a useful label for the four products in a binomial times a binomial, and students who already know it should keep it. The problem starts when a student treats it as the rule for multiplying polynomials and then meets a trinomial. Introducing the box or the vertical layout alongside FOIL, on the same example, is what stops that from happening.

Which method should a struggling student use?

The box. It is the only one of the four where a skipped product leaves a visible empty cell, which means the student can find their own error instead of waiting for the paper to come back. Once the products are reliable, the vertical layout is faster for long polynomials.

What grade level is this set for?

Algebra I, typically grades 8 to 10, and as an Algebra II review before rational expressions or polynomial division. No calculator is required.

Can I use these in my classroom?

Yes. Print, photocopy, and distribute them to your own students freely. Please do not repost the PDF files on another site or resell them. A link back to this page is appreciated if you share them with colleagues.

What comes after polynomial operations?

Factoring, which is these same products run backwards, then polynomial division and graphing. The factoring polynomials flowchart guide is the direct next step.

More polynomial practice

Questions were prepared from previous class notes and assigned homework, and every answer was checked for accuracy before publishing. Published August 7, 2026. Last updated August 7, 2026.