Simplify polynomial expressions by adding, subtracting, and multiplying

Ontario Grade 11 Mathematics

Characteristics of Functions

Adding, Subtracting, and Multiplying Polynomials

Polynomials show up throughout MCR3U — in function rules, in modelling situations, and in manipulating expressions before graphing or solving. Before you can work confidently with quadratic or higher-degree functions, you need to be able to add, subtract, and multiply polynomial expressions quickly and accurately. This lesson builds directly on the Grade 10 skill of expanding brackets and collecting like terms, and then pushes further into multiplying larger polynomials. Every new idea is introduced with a plain-language explanation, a concrete numerical check, and then the general symbolic form.

What you will learn

Identify like terms in a polynomial expression and combine them correctly.
Add and subtract polynomial expressions by collecting like terms.
Multiply a polynomial by a monomial using the distributive property.
Multiply two binomials or a binomial by a trinomial and simplify the result.

Prerequisite Bridge: What Is a Polynomial?

A polynomial is an expression made up of one or more terms. Each term is a number, a variable, or a product of a number and one or more variables raised to whole-number powers. Examples include

3x^2

,

-7x

, and

5

. A term like

\frac{1}{x}

or

x^{-2}

is not a polynomial term because the variable is in the denominator or has a negative exponent.

The degree of a term is the sum of the exponents on its variables. For example,

4x^3y

has degree

3 + 1 = 4

. The degree of the whole polynomial is the highest degree among all its terms. A polynomial with one term is called a monomial, two terms a binomial, and three terms a trinomial.

Like terms share exactly the same variable part — the same variables raised to the same powers. So

5x^2

and

-2x^2

are like terms, but

5x^2

and

5x

are not, because the exponents differ. You can only combine like terms.

A polynomial term has variables with whole-number (non-negative integer) exponents only.
The degree of a term is the total of all exponents on its variables.
Like terms have identical variable parts and can be combined by adding or subtracting their coefficients.

Adding and Subtracting Polynomials

Adding polynomials means writing both expressions together and then collecting like terms. The key idea is that only like terms combine — you add or subtract their coefficients and keep the variable part unchanged. Think of

3x^2 + 5x^2

the same way you think of

3

apples plus

5

apples: the result is

8

of the same thing, so

8x^2

.

Subtracting a polynomial requires an extra step: you must distribute the negative sign to every term inside the second set of brackets before collecting like terms. For example,

(6x^2 - 4x + 1) - (2x^2 + 3x - 5)

becomes

6x^2 - 4x + 1 - 2x^2 - 3x + 5

after the negative sign is applied. Notice how every sign inside the second bracket flips. Skipping this step is the most common source of errors in subtraction.

Once all brackets are removed, group like terms together — it helps to rearrange them so that terms with the same degree sit beside each other — and then combine. Write your answer in descending order of degree (highest power first), which is the conventional form for polynomials.

(a + b) - (c + d) = a + b - c - d

Add polynomials by grouping and combining like terms.
Subtract by first distributing the negative sign to every term in the bracket being subtracted.
Write the simplified result in descending order of degree.

Multiplying a Polynomial by a Monomial

The distributive property says that a factor outside a bracket multiplies every single term inside the bracket. This applies whether the factor is a number, a variable, or a product of both. For instance,

3x(2x^2 - 5x + 4)

means

3x

multiplies each of the three terms separately.

When you multiply two powers that share the same base, you add the exponents. This is the exponent rule

x^m \cdot x^n = x^{m+n}

, which you met in Grade 9 and 10. So

3x \cdot 2x^2 = 6x^3

, because

3 \cdot 2 = 6

and

x^1 \cdot x^2 = x^3

.

After distributing, check whether any like terms appeared. If the monomial multiplied a trinomial, you will have three terms; usually none of them are like terms at this stage, but always check before declaring the expression simplified.

a(b + c + d) = ab + ac + ad

Distribute the monomial by multiplying it with every term inside the bracket.
Multiply coefficients together and add exponents of like bases.
Check for any like terms in the result before finishing.

Multiplying Two Polynomials

When multiplying two multi-term polynomials — such as a binomial times a binomial, or a binomial times a trinomial — every term in the first polynomial must multiply every term in the second polynomial. A reliable method is to distribute each term of the first polynomial across the entire second polynomial, one term at a time, and then collect any like terms in the result.

For two binomials, this produces four partial products before simplifying. You may recognise the pattern sometimes called FOIL (First, Outer, Inner, Last) from Grade 10; it is simply a memory aid for applying the distributive property twice. However, FOIL only works for binomial × binomial. For larger products — such as a binomial times a trinomial — the full distribute-each-term approach always works and produces six partial products before simplifying.

After forming all partial products, look carefully for like terms. Typically the middle-degree terms combine. Always multiply coefficients, add matching exponents, and arrange the final answer in descending order of degree. A numerical check — substituting a simple value like

x = 1

into both the original product and your simplified answer — is a fast way to catch errors.

(a + b)(c + d) = ac + ad + bc + bd

Every term in the first polynomial multiplies every term in the second polynomial.
Collect and combine like terms in the result.
Arrange the final answer in descending order of degree.
A substitution check (e.g., $x = 1$ ) can quickly verify your answer.

Putting It All Together: Mixed Operations

Some expressions require more than one operation in sequence. For example, you might need to expand two products and then subtract one result from the other. The order of operations still applies: work inside brackets first, then multiply, then add or subtract.

A useful habit is to expand each product separately, writing out all terms, before combining everything. Trying to do too many steps mentally at once is where sign errors and missing terms creep in. Label each partial result clearly so you can track what you are adding or subtracting.

After any simplification, double-check by choosing a specific value — say

x = 2

— and evaluating the original expression and your simplified expression. If both give the same number, the simplification is correct. This substitution strategy does not prove that two expressions are always equal (you would need algebra for that), but it is an excellent way to catch arithmetic mistakes before moving on.

Follow the order of operations: expand brackets before adding or subtracting.
Expand each product separately to reduce errors.
Use substitution of a specific value to verify your simplified expression.

Identifying Like Terms at a Glance

Term	Variable Part	Like Terms With
$5x^2$	$x^2$	$-2x^2$ , 9x^2
$-3xy$	$xy$	$7xy$ , -xy
$4x$	$x$	$x$ , -6x
$8$	none (constant)	$-3$ , 11
$2x^2y$	$x^2y$	$-5x^2y$

Worked example

Adding and Subtracting Two Trinomials

Simplify:

(4x^2 - 3x + 7) + (x^2 + 5x - 2) - (2x^2 - x - 4)

Remove all brackets
The first two sets of brackets have a positive sign in front, so those terms stay as they are. The third set has a negative sign in front, so every term inside it changes sign. Write out all terms without brackets.
$4x^2 - 3x + 7 + x^2 + 5x - 2 - 2x^2 + x + 4$
Group like terms
Rearrange so that terms with $x^2$ , terms with $x$ , and constant terms are beside each other. This makes it easier to see what combines.
$(4x^2 + x^2 - 2x^2) + (-3x + 5x + x) + (7 - 2 + 4)$
Combine each group
Add the coefficients within each group: $4 + 1 - 2 = 3$ for the $x^2$ terms, $-3 + 5 + 1 = 3$ for the $x$ terms, and $7 - 2 + 4 = 9$ for the constants.
$3x^2 + 3x + 9$

Answer:

3x^2 + 3x + 9

Check: Substitute

x = 1

into the original:

(4 - 3 + 7) + (1 + 5 - 2) - (2 - 1 - 4) = 8 + 4 - (-3) = 8 + 4 + 3 = 15

. Substitute

x = 1

into the answer:

3(1)^2 + 3(1) + 9 = 3 + 3 + 9 = 15

. ✓

Worked example

Multiplying a Binomial by a Trinomial

Expand and simplify:

(2x - 3)(x^2 + 4x - 5)

Distribute the first term of the binomial
Multiply $2x$ by every term in the trinomial. Use $x^m \cdot x^n = x^{m+n}$ for the variable parts and multiply the coefficients normally.
$2x · x^2 + 2x · 4x + 2x · (-5) = 2x^3 + 8x^2 - 10x$
Distribute the second term of the binomial
Now multiply $-3$ by every term in the trinomial. Be careful with the signs: a negative times a negative gives a positive.
$(-3) · x^2 + (-3) · 4x + (-3) · (-5) = -3x^2 - 12x + 15$
Write all six partial products together
Combine the results of both distributions into one expression before looking for like terms.
$2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15$
Collect and combine like terms
The only like terms here are the $x^2$ terms ( $8x^2$ and $-3x^2$ ) and the $x$ terms ( $-10x$ and $-12x$ ). Combine their coefficients: $8 - 3 = 5$ and $-10 - 12 = -22$ .
$2x^3 + 5x^2 - 22x + 15$

Answer:

2x^3 + 5x^2 - 22x + 15

Check: Substitute

x = 1

into the original:

(2(1) - 3)((1)^2 + 4(1) - 5) = (2 - 3)(1 + 4 - 5) = (-1)(0) = 0

. Substitute

x = 1

into the answer:

2(1)^3 + 5(1)^2 - 22(1) + 15 = 2 + 5 - 22 + 15 = 0

. ✓

Common mistakes and how to avoid them

When subtracting a polynomial, forgetting to flip the sign on every term inside the bracket — for example, writing

(5x - 3) - (2x - 4) = 5x - 3 - 2x - 4

instead of correctly getting

5x - 3 - 2x + 4

.

Correction: Think of the subtraction sign as multiplying every term inside the bracket by

-1

. Change every sign inside before collecting like terms.

Adding exponents when adding like terms — for example, writing

3x^2 + 4x^2 = 7x^4

instead of

7x^2

.

Correction: You only add exponents when multiplying powers of the same base, not when adding terms. Adding like terms changes only the coefficient:

3x^2 + 4x^2 = 7x^2

.

Multiplying coefficients but forgetting to add exponents — for example, writing

3x \cdot 2x^2 = 6x^2

instead of

6x^3

.

Correction: When multiplying, handle coefficients and variable parts separately. Multiply the coefficients (

3 \times 2 = 6

) and add the exponents (

x^1 \cdot x^2 = x^3

), giving

6x^3

.

When multiplying two binomials, only forming two products instead of four — for example, treating

(x + 3)(x + 5)

as just

x^2 + 15

.

Correction: Every term in the first binomial must multiply every term in the second.

(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15

.

Combining unlike terms — for example, adding

5x^2 + 3x = 8x^3

or

4x^2 + 2 = 6x^2

.

Correction: Only terms with identical variable parts can be combined.

5x^2

and

3x

have different variable parts and must remain as separate terms.

Lesson summary

A polynomial is an expression with terms that have variables raised to whole-number powers. Like terms share identical variable parts and can be combined.
To add polynomials, remove brackets and collect like terms by adding their coefficients.
To subtract a polynomial, distribute the negative sign to every term in the bracket before collecting like terms.
To multiply a polynomial by a monomial, use the distributive property: multiply the monomial by each term, multiplying coefficients and adding exponents of like bases.
To multiply two multi-term polynomials, distribute each term of the first polynomial across every term of the second, then collect like terms and write the result in descending order of degree.
A quick substitution check — evaluating both the original and simplified expressions at a chosen value of $x$ — is a reliable way to verify your answer.

Check your understanding

Question 1

Which of the following is the correct result of

(3x^2 - 2x + 1) - (x^2 + 4x - 6)

?

$2x^2 - 6x - 5$
$2x^2 + 2x + 7$
$2x^2 - 6x + 7$
$4x^2 - 6x - 5$

Show answer and explanation

2x^2 - 6x + 7

Distribute the negative sign:

3x^2 - 2x + 1 - x^2 - 4x + 6

. Combine

x^2

terms:

3 - 1 = 2

, so

2x^2

. Combine

x

terms:

-2 - 4 = -6

, so

-6x

. Combine constants:

1 + 6 = 7

. The result is

2x^2 - 6x + 7

.

Question 2

What is the result of multiplying

-3x^2(4x - 5)

?

$-12x^2 + 15x$
$-12x^3 - 15x^2$
$-12x^3 + 15x^2$
$12x^3 - 15x^2$

Show answer and explanation

-12x^3 + 15x^2

-3x^2 \cdot 4x = -12x^3

and

-3x^2 \cdot (-5) = +15x^2

. A negative times a negative is positive, giving

-12x^3 + 15x^2

.

Question 3

Expand and simplify

(x + 4)(x - 6)

.

$x^2 - 2x - 24$
$x^2 + 2x - 24$
$x^2 - 24$
$x^2 - 2x + 24$

Show answer and explanation

x^2 - 2x - 24

Distribute:

x \cdot x = x^2

;

x \cdot (-6) = -6x

;

4 \cdot x = 4x

;

4 \cdot (-6) = -24

. Combine like terms:

-6x + 4x = -2x

. Result:

x^2 - 2x - 24

.

Question 4

Which expression is NOT simplified correctly?

$2x^2 + 5x^2 = 7x^2$
$3x \cdot 4x^2 = 12x^3$
$6x^3 - 2x^3 = 4x^6$
$7x - 3x = 4x$

Show answer and explanation

6x^3 - 2x^3 = 4x^6

6x^3 - 2x^3

should equal

4x^3

, not

4x^6

. Subtracting like terms changes only the coefficient; the exponent stays the same. The other three options are all correct.

Key terms

Polynomial: An algebraic expression with one or more terms, where each variable is raised to a non-negative whole-number exponent.
Term: A single number, variable, or product of numbers and variables in an expression. Terms are separated by addition or subtraction signs.
Coefficient: The numerical factor in a term. In $-7x^2$ , the coefficient is $-7$ .
Degree (of a term): The sum of all exponents of the variables in a single term. The term $5x^2y$ has degree $2 + 1 = 3$ .
Like terms: Terms that have exactly the same variable part (same variables raised to the same powers). Only like terms can be combined by adding or subtracting.
Distributive property: The rule that $a(b + c) = ab + ac$ . A factor outside a bracket multiplies every term inside the bracket.
Monomial: A polynomial with exactly one term, such as $-4x^3$ or $7$ .
Binomial: A polynomial with exactly two terms, such as $3x - 5$ .

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A3.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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