Teaching unit · Algebra 1 (commonly taken Grades 8 to 10)

Adding, subtracting and multiplying polynomials

Polynomials behave like a number system: combine like terms to add and subtract, distribute to multiply

About three to four lessons of 45 to 60 minutes

Start here · hook

Polynomials add, subtract and multiply just like whole numbers

Add two whole numbers and you always get a whole number back. Multiply two whole numbers and you always get a whole number back. The integers are 'closed' under those operations, they never kick you out of the system. Polynomials work exactly the same way: add, subtract or multiply two polynomials and the result is always another polynomial.

That single idea is the whole point of this unit. Combining like terms IS polynomial addition and subtraction, and the distributive property IS polynomial multiplication, whether you are multiplying a single term into a bracket, two binomials together (FOIL), or a binomial into a trinomial to reach a cubic expression.

Total volume of two differently-sized boxeseach side length is an expression in x; adding volumes adds polynomials
Profit = revenue - costwhen both revenue and cost are polynomial expressions in the number sold, x
The area of a rectangle with algebraic side lengths(2x + 3)(4x - 5): multiplying two binomials to get one polynomial
Volume of a box built from a flat sheetcutting squares of side x from each corner gives a CUBIC volume polynomial

Learning objective

What students will be able to do

Students will add and subtract polynomials by combining like terms, and multiply polynomials (a monomial by a polynomial, a binomial by a binomial including non-monic factors, and a binomial by a trinomial) by applying the distributive property and combining like terms.

Success criteria

I can add or subtract two polynomials by combining like terms, matching each term to its degree.
I can distribute a monomial across a polynomial.
I can multiply two binomials, including when neither has a leading coefficient of 1.
I can multiply a binomial by a trinomial to reach a cubic expression, combining every like term correctly.

Curriculum anchor

Standards this unit teaches

HSA-APR.A.1Common Core (US)
Add, subtract and multiply polynomials
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Polynomial: an expression built from terms with whole-number exponents, such as 3x² - 2x + 5
Term: one piece of a polynomial, a coefficient times a power of the variable (or a constant)
Like terms: terms with exactly the same variable raised to exactly the same power, e.g. 3x² and -5x²
Degree: the highest power of the variable in a polynomial; a degree-3 (cubic) polynomial's highest term is an x³ term
Monomial / binomial / trinomial: a polynomial with one, two, or three terms
Non-monic: having a leading coefficient other than 1, e.g. the 2 in 2x + 3

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Adding and subtracting polynomials: combining like terms

Concrete

Adding or subtracting two polynomials only ever combines LIKE terms, terms with the same variable raised to the same power. An x² term can only combine with another x² term, never with an x term or a constant, the same way you would never add '3 apples' and '2 oranges' together as a single count.

Line the two polynomials up by degree (x², x, constant, and so on), then add or subtract the coefficients of each matching pair. Any degree missing from one polynomial just contributes 0 for that term. Subtracting a polynomial means every one of ITS terms flips sign before combining, exactly like subtracting a bracketed expression in ordinary algebra.

Worked example

Simplify: (3x² - 2x + 5) + (x² + 4x - 7)

Match terms by degree: x² with x², x with x, constant with constant.
x² terms: 3 + 1 = 4, giving 4x².
x terms: -2 + 4 = 2, giving 2x.
constant terms: 5 + (-7) = -2.

Answer: 4x² + 2x - 2

Check for understanding, ask

Why can 3x² and 5x not be combined into a single term?
When subtracting (x² + 4x - 7), what happens to each of its three terms' signs?
What degree is the result of adding a degree-2 polynomial to a degree-3 polynomial?

2. Multiplying polynomials: the distributive property, including non-monic binomials

Pictorial

Multiplying a single term into a bracket distributes that term across every term inside: a(b + c) = ab + ac. Multiplying two binomials is the same idea done twice, every term in the first bracket distributes across every term in the second (the FOIL pattern: First, Outer, Inner, Last), then any like terms produced combine.

Unlike the factorising unit above, which only ever expands MONIC binomials like (x + a)(x + b), a real polynomial multiplication can have a leading coefficient greater than 1 on either or both factors, for example (2x + 3)(4x - 5). The process is identical, just with more arithmetic in each of the four products.

y = 8x² + 2x - 15, the product (2x + 3)(4x - 5) expanded below. Plotting it confirms it is a single parabola, exactly what multiplying two linear factors should produce.

Worked example

Expand: (2x + 3)(4x - 5)

First: (2x)(4x) = 8x².
Outer: (2x)(-5) = -10x.
Inner: (3)(4x) = 12x.
Last: (3)(-5) = -15.
Combine the outer and inner like terms: -10x + 12x = 2x.

Answer: 8x² + 2x - 15

Check for understanding, ask

Why does FOIL produce exactly four products for two binomials?
What would change in the working if the second binomial were (4x + 5) instead of (4x - 5)?
Is (2x + 3)(4x - 5) the same as (4x - 5)(2x + 3)? Why?

3. Multiplying to a cubic: a binomial times a trinomial

Abstract

The same distributive idea scales up to any size polynomial: every term in the first factor multiplies every term in the second factor, then like terms combine. Multiplying a binomial (2 terms) by a trinomial (3 terms) produces 6 products before combining, and reaches a genuine CUBIC (degree-3) result, one whole degree beyond any quadratic.

There is no new rule here, only more bookkeeping: track every one of the 6 products by degree (adding exponents, since $x^{m}$ . $x^{n}$ = $x^{m+n}$ ), then combine like terms exactly as in section 1.

y = x³ - x² - 2x + 8, the product (x + 2)(x² - 3x + 4) expanded below. Unlike a parabola, a cubic can rise, dip and rise again, visible in the wiggle between x = -1 and x = 2.

Worked example

Expand: (x + 2)(x² - 3x + 4)

Distribute x across the trinomial: x(x²) + x(-3x) + x(4) = x³ - 3x² + 4x.
Distribute 2 across the trinomial: 2(x²) + 2(-3x) + 2(4) = 2x² - 6x + 8.
Add the two results and combine like terms: x³ + (-3x² + 2x²) + (4x - 6x) + 8.

Answer: x³ - x² - 2x + 8

Check for understanding, ask

How many individual products are there before combining like terms, for a binomial times a trinomial?
Why does the result have degree 3 when the two factors have degree 1 and degree 2?
Could a binomial times a trinomial ever produce a degree-2 result? Why or why not?

Watch for

Common misconceptions and how to address them

Misconception3x² + 5x combines into 8x² or 8x.

Why it happens: Students over-generalise 'just add the numbers' from combining true like terms, forgetting the powers must match too.

How to address it: Only terms with the SAME variable raised to the SAME power can combine. 3x² and 5x have different powers (2 and 1), so 3x² + 5x is already fully simplified, it cannot be combined further.

MisconceptionSubtracting a polynomial only flips the sign of its FIRST term.

Why it happens: Students treat the subtraction sign like it belongs only to the term right after it, the same slip that shows up subtracting an ordinary bracketed expression.

How to address it: Subtracting (a + b - c) flips the sign of EVERY term inside: -(a + b - c) = -a - b + c. Rewrite a subtraction as 'add the opposite of every term' before combining.

MisconceptionFOIL only works when both binomials start with a plain x (a leading coefficient of 1).

Why it happens: Most first examples students see are monic, like (x + a)(x + b), so the pattern feels tied to that specific case.

How to address it: FOIL (or full distribution) works for ANY two binomials, coefficients included: (2x + 3)(4x - 5) uses the exact same First/Outer/Inner/Last process, just with bigger numbers to multiply.

MisconceptionMultiplying a binomial by a trinomial should give a degree-2 (quadratic) result, since 'multiplying makes things bigger, not the degree'.

Why it happens: Students conflate the SIZE of a product with its DEGREE, and have mostly seen binomial x binomial (degree 1 + degree 1 = degree 2) so far.

How to address it: The degree of a product is the SUM of the degrees of the factors (a rule that follows from $x^{m}$ . $x^{n}$ = $x^{m+n}$ on the leading terms). A degree-1 binomial times a degree-2 trinomial always gives a degree-3 (cubic) result.

Do it together

Guided practice (with answers)

1. Simplify: (5x² + 3x - 1) + (2x² - 6x + 4)
Answer: 7x² - 3x + 3, because x² terms 5+2=7, x terms 3-6=-3, constants -1+4=3.
2. Simplify: (4x² - x + 6) - (x² + 5x - 2)
Answer: 3x² - 6x + 8, because subtracting flips every term of the second polynomial: (4-1)x² + (-1-5)x + (6-(-2)).
3. Expand: 3x(2x² - 4x + 1)
Answer: 6x³ - 12x² + 3x, distributing 3x across every term.
4. Expand: (3x - 2)(2x + 7)
Answer: 6x² + 17x - 14, because First 6x², Outer 21x, Inner -4x, Last -14, and 21x - 4x = 17x.
5. Expand: (x - 1)(x² + x + 1)
Answer: x³ - 1, because x(x²+x+1) = x³+x²+x and -1(x²+x+1) = -x²-x-1, and every middle term cancels.

On their own

Independent practice worksheets

Practise adding, subtracting and multiplying polynomials with computed, never-wrong answer keys.

Algebra 1: adding and subtracting polynomialsthe core skill from section 1, freshly generated each time Algebra 1: multiplying polynomialsdistributing, non-monic binomials, and a binomial times a trinomial from sections 2 and 3 Year 9 quadratics (monic factoring) worksheetrevisit the monic-only expanding this unit builds beyond

Reach every student

Differentiation

Support

Start adding/subtracting with two polynomials that already list terms in the same order and degree, before introducing missing degrees.
Use a two-column table (one column per polynomial, rows for each degree) as scaffolding while combining like terms.
Begin multiplication with a monomial times a binomial (2 products) before a full binomial times binomial (4 products) or binomial times trinomial (6 products).
Colour-code First/Outer/Inner/Last on the board for the first few binomial products.

Extension

Multiply two trinomials together (9 products) and check the result's degree matches the sum of the factors' degrees.
Explore (x + a)(x - a), noticing the middle terms always cancel (a difference of squares), and predict the pattern before expanding a new example.
Ask students to write a word problem whose solution requires multiplying two polynomial expressions, such as a box volume.
Verify a multiplication by substituting a specific number for x into both the original product and the expanded answer, confirming they match.

Check it stuck

Assessment: exit ticket

A short exit ticket sampling adding, subtracting and multiplying polynomials.

1. Simplify: (2x² + x - 5) - (x² - 3x + 1)
Answer: x² + 4x - 6, because subtracting flips every term of the second polynomial before combining.
2. Expand: (5x - 2)(x + 4)
Answer: 5x² + 18x - 8, because First 5x², Outer 20x, Inner -2x, Last -8, and 20x - 2x = 18x.
3. What degree does multiplying a degree-1 binomial by a degree-2 trinomial produce, and why?
Answer: Degree 3, because the degree of a product is the sum of the factors' degrees (1 + 2 = 3).

For the teacher

Teacher notes and timings

Rough timing across three to four lessons: Lesson 1 adding/subtracting (section 1), Lesson 2 distributing and non-monic FOIL (section 2), Lesson 3 binomial x trinomial (section 3), Lesson 4 mixed practice plus the exit ticket.
Prior knowledge: the factoring-based quadratics unit (year-9-quadratics-expand-factorise-solve, linked above) explicitly only ever expands MONIC binomials (x + a)(x + b). This unit deliberately goes further: non-monic binomials (leading coefficient greater than 1) and a binomial times a trinomial reaching a genuine cubic, neither of which exists anywhere else on the site.
Language to repeat: 'like terms' need the SAME variable AND the SAME power; the degree of a product is the SUM of the factors' degrees; subtracting a polynomial flips every one of its terms' signs, not just the first.
Curriculum note: Common Core HSA-APR.A.1 frames polynomials as a system closed under addition, subtraction and multiplication, analogous to the integers; making that analogy explicit in the hook helps students see WHY the procedures always work, not just THAT they work. Verified live at thecorestandards.org/Math/Content/HSA/APR/A/1/.

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