What is an algebraic expression in the context of IB MYP Mathematics?

In the IB MYP Mathematics curriculum, an algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions do not have an equality sign, distinguishing them from equations. For example, 3x + 5 is an algebraic expression where 'x' is a variable.

How do you simplify an algebraic expression?

To simplify an algebraic expression, you combine like terms, which are terms that have the same variable raised to the same power. For instance, in the expression 2x + 3x - 4, you can combine 2x and 3x to get 5x, resulting in the simplified expression 5x - 4. Simplifying helps in making expressions easier to work with, especially in solving equations.

What is the difference between an algebraic expression and an equation?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations, but it does not have an equality sign. An equation, on the other hand, is a statement that two expressions are equal, indicated by an equality sign. For example, 2x + 3 is an expression, while 2x + 3 = 7 is an equation.

How do you evaluate an algebraic expression?

To evaluate an algebraic expression, you substitute the values of the variables into the expression and then perform the arithmetic operations. For example, if you have the expression 3x + 2 and you need to evaluate it for x = 4, you substitute 4 for x, resulting in 3(4) + 2 = 12 + 2 = 14.

What are the common types of algebraic expressions encountered in IB MYP Mathematics?

In the IB MYP Mathematics curriculum, students commonly encounter several types of algebraic expressions, including monomials (single term, e.g., 5x), binomials (two terms, e.g., x + 2), and polynomials (multiple terms, e.g., 3x^2 + 2x + 1). Understanding these types helps in performing operations like addition, subtraction, and multiplication of expressions.

Why is "Combining $3x^2$ with $3x$." a common mistake in Algebraic Expressions?

Why it happens: Same letter — treated as like. How to avoid it: Like terms need the same POWER too. 3x^2 and 3x are NOT like; they cannot be combined.

Why is "Distributing $-(2x + 3) = -2x + 3$." a common mistake in Algebraic Expressions?

Why it happens: Forgetting the negative applies to BOTH terms. How to avoid it: -(2x + 3) = -2x - 3. Distribute the minus to EVERY term.

Why is "Saying $(x + 3)^2 = x^2 + 9$." a common mistake in Algebraic Expressions?

Why it happens: Squaring each term separately. How to avoid it: (x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9. You MUST expand with FOIL or use (a+b)^2 = a^2 + 2ab + b^2.

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Detailed Study Notes

Detailed notes on Algebra for IB MYP Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Algebraic Expressions — IB MYP Mathematics (Extended): variables, terms, and the four operations

Algebra generalises arithmetic — letters stand for any number. This MYP Mathematics Extended note covers terms, like terms, expansion, and substitution.

At a glance

A VARIABLE is a letter standing for any number (e.g. $x$ , $y$ , $n$ ).
A TERM is a number, a letter, or a product of these (e.g. $3x$ , $-2xy^2$ ).
LIKE TERMS have the SAME letters with the SAME powers — can be added/subtracted.
EXPAND brackets: $a(b+c) = ab + ac$ .
EXPAND two brackets: FOIL — First, Outer, Inner, Last.
SUBSTITUTE: replace each letter with its value (using brackets!).

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Identify terms and coefficients.
MYP Mathematics A — Simplify by collecting like terms.
MYP Mathematics A — Expand single and double brackets.
MYP Mathematics C — Substitute values into algebraic expressions.

Terms and like terms

Combine what's the same; keep what's different.

An algebraic expression is built from terms joined by $+$ or $-$ signs:

$3x^2 - 4xy + 7y - 2.$

Each term has:

A coefficient (the number in front, e.g. 3 in $3x^2$ ).
Letters with powers (e.g. $x^2$ , $xy$ , $y$ ).
A constant term may have no letters (e.g. $-2$ ).

LIKE TERMS have the SAME letters with the SAME powers. Coefficient differences are fine.

$3x$ and $7x$ → like.
$5xy$ and $-2xy$ → like.
$4x^2$ and $4x$ → NOT like ( $x^2 \ne x$ ).
$3xy$ and $3yx$ → like ( $xy = yx$ ).

You can ADD or SUBTRACT like terms by combining their coefficients:

$5x + 3x = 8x.$ $7x^2y - 4x^2y = 3x^2y.$

You CANNOT combine UNLIKE terms. $3x + 5y$ stays as $3x + 5y$ .

Worked example. Simplify $4a + 7b - 2a + 5b - 3$ .

Group like: $(4a - 2a) + (7b + 5b) - 3$ .
Combine: $2a + 12b - 3$ .

Like terms: same letters, same powers.
Add coefficients of like terms.
Unlike terms stay separate.

Expanding brackets

Distribute the multiplication.

Single bracket: multiply everything inside by what's outside.

$a(b + c) = ab + ac.$

Worked example. Expand $3(2x - 5y + 4)$ .

$3 \times 2x = 6x$ .
$3 \times (-5y) = -15y$ .
$3 \times 4 = 12$ .
Result: $6x - 15y + 12$ .

Two brackets (FOIL). $(a + b)(c + d) = ac + ad + bc + bd$ .

The acronym FOIL (First, Outer, Inner, Last) helps you remember to multiply every term in the first bracket by every term in the second.

Worked example. Expand $(x + 3)(x + 5)$ .

F: $x \times x = x^2$ .
O: $x \times 5 = 5x$ .
I: $3 \times x = 3x$ .
L: $3 \times 5 = 15$ .
Sum: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$ .

Worked example (Extended). Expand $(2x - 3)(x + 4)$ .

F: $2x \times x = 2x^2$ .
O: $2x \times 4 = 8x$ .
I: $-3 \times x = -3x$ .
L: $-3 \times 4 = -12$ .
Sum: $2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$ .

Three special identities to memorise:

$(a+b)^2 = a^2 + 2ab + b^2$ .
$(a-b)^2 = a^2 - 2ab + b^2$ .
$(a+b)(a-b) = a^2 - b^2$ (difference of squares).

These show up CONSTANTLY in factorisation and algebra problems.

Single bracket: distribute the multiplier.
Double bracket: FOIL — every term from one × every term from the other.
$(a+b)^2 = a^2 + 2ab + b^2$ .
$(a-b)(a+b) = a^2 - b^2$ .

Substitution and evaluating

Replace letters with values — use brackets.

Substitution means replacing each letter with its given value, then evaluating.

Worked example. Find the value of $3x^2 - 2x + 5$ when $x = -4$ .

ALWAYS use brackets around the substituted value:

$3(-4)^2 - 2(-4) + 5$ .
BIDMAS: $(-4)^2 = 16$ .
$3 \times 16 - 2 \times (-4) + 5 = 48 + 8 + 5 = 61$ .

(Without brackets, $-4^2$ would be $-16$ — wrong by sign.)

Worked example. Find the value of $\dfrac{a^2 - b^2}{2a + b}$ when $a = 5$ and $b = 3$ .

Top: $5^2 - 3^2 = 25 - 9 = 16$ .
Bottom: $2(5) + 3 = 13$ .
Result: $\dfrac{16}{13}$ .

Multi-step. Sometimes you need to substitute into one formula then into another:

Volume of a cone: $V = \tfrac{1}{3}\pi r^2 h$ . Find $V$ when $r = 3$ and $h = 5$ .

$V = \tfrac{1}{3} \times \pi \times 3^2 \times 5 = \tfrac{1}{3} \times \pi \times 9 \times 5 = 15\pi \approx 47.1$ .

A simple rule: at the start of any substitution problem, REWRITE the expression with brackets around each substituted value. This single habit removes most sign errors.

Always BRACKET substituted values.
Apply BIDMAS.
Negative values especially need brackets.

Quick recap

Like terms: same letters, same powers. Add coefficients.
Single bracket: distribute.
Double bracket: FOIL.
$(a+b)^2 = a^2 + 2ab + b^2$ . $(a-b)(a+b) = a^2 - b^2$ .
Substitution: BRACKET each value.

Memorise this

Verbatim phrases and definitions MYP criterion-A markschemes credit.

$(a+b)^2 = a^2 + 2ab + b^2$
$(a-b)^2 = a^2 - 2ab + b^2$
$(a+b)(a-b) = a^2 - b^2$
$a(b+c) = ab + ac$
FOIL: First, Outer, Inner, Last

How it’s examined

Criterion A: simplify and expand. Criterion C: clear bracket-by-bracket working. Criterion D: substitute values into formulae from real contexts.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

Take this whole topic with you

Step-by-step worked examples — Algebraic Expressions

Step-by-step solutions to past-paper-style questions on algebraic expressions, written exactly the way a tutor would explain them at the board.

1Collect like terms
Getting started• simplify
Question
Simplify $5a + 3b - 2a + 7b - 4$ .
Step-by-step solution
1. Step 1
  Group like terms: $(5a - 2a) + (3b + 7b) - 4$ .
2. Step 2
  Combine: $3a + 10b - 4$ .
Answer
$3a + 10b - 4$
2Expand two brackets
Building confidence• expand
Question
Expand and simplify $(x - 4)(x + 7)$ .
Step-by-step solution
1. Step 1
  FOIL: First $x \times x = x^2$ . Outer $x \times 7 = 7x$ . Inner $-4 \times x = -4x$ . Last $-4 \times 7 = -28$ .
2. Step 2
  Sum: $x^2 + 7x - 4x - 28 = x^2 + 3x - 28$ .
Answer
$x^2 + 3x - 28$
3Use a special identity
Building confidence• identity
Question
Expand $(3x - 2)^2$ without using FOIL.
Step-by-step solution
1. Step 1
  Use $(a-b)^2 = a^2 - 2ab + b^2$ with $a = 3x$ and $b = 2$ .
2. Step 2
  $a^2 = 9x^2$ , $2ab = 2 \times 3x \times 2 = 12x$ , $b^2 = 4$ .
3. Step 3
  Result: $9x^2 - 12x + 4$ .
Answer
$9x^2 - 12x + 4$
4Substitute a negative
Stretch• substitution
Question
Find the value of $2x^2 - 5x + 3$ when $x = -3$ .
Step-by-step solution
1. Step 1
  Substitute WITH brackets: $2(-3)^2 - 5(-3) + 3$ .
2. Step 2
  BIDMAS: $(-3)^2 = 9$ . Expression: $2(9) - 5(-3) + 3$ .
3. Step 3
  Multiplications: $18 + 15 + 3$ .
4. Step 4
  Sum: $36$ .
Answer
$36$

Key Definitions and Keywords — Algebraic Expressions

Definitions to memorise and the exact keywords mark schemes credit for algebraic expressions answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Variable
Examiner keyword
A letter standing for an unknown or generic number.
Term
Examiner keyword
A single number, a single letter, or a product of these.
Coefficient
Examiner keyword
The numerical part of a term. The coefficient of $3x$ is 3.
Like terms
Examiner keyword
Terms with the same letters at the same powers. Can be combined by adding coefficients.
Expand
Remove brackets by multiplying out.

Common Mistakes and Misconceptions — Algebraic Expressions

The traps other students keep falling into on algebraic expressions questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Combining $3x^2$ with $3x$ .
Why it happens
Same letter — treated as like.
How to avoid it
Like terms need the same POWER too. $3x^2$ and $3x$ are NOT like; they cannot be combined.
✕Distributing $-(2x + 3) = -2x + 3$ .
Why it happens
Forgetting the negative applies to BOTH terms.
How to avoid it
$-(2x + 3) = -2x - 3$ . Distribute the minus to EVERY term.
✕Saying $(x + 3)^2 = x^2 + 9$ .
Why it happens
Squaring each term separately.
How to avoid it
$(x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9$ . You MUST expand with FOIL or use $(a+b)^2 = a^2 + 2ab + b^2$ .

Algebraic Expressions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

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Algebraic Expressions

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Short Notes - Algebraic Expressions

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Detailed Study Notes

Detailed notes on Algebra for IB MYP Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Algebraic Expressions — IB MYP Mathematics (Extended): variables, terms, and the four operations

Algebra generalises arithmetic — letters stand for any number. This MYP Mathematics Extended note covers terms, like terms, expansion, and substitution.

At a glance

A VARIABLE is a letter standing for any number (e.g. $x$ , $y$ , $n$ ).
A TERM is a number, a letter, or a product of these (e.g. $3x$ , $-2xy^2$ ).
LIKE TERMS have the SAME letters with the SAME powers — can be added/subtracted.
EXPAND brackets: $a(b+c) = ab + ac$ .
EXPAND two brackets: FOIL — First, Outer, Inner, Last.
SUBSTITUTE: replace each letter with its value (using brackets!).

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Identify terms and coefficients.
MYP Mathematics A — Simplify by collecting like terms.
MYP Mathematics A — Expand single and double brackets.
MYP Mathematics C — Substitute values into algebraic expressions.

Terms and like terms

Combine what's the same; keep what's different.

An algebraic expression is built from terms joined by $+$ or $-$ signs:

$3x^2 - 4xy + 7y - 2.$

Each term has:

A coefficient (the number in front, e.g. 3 in $3x^2$ ).
Letters with powers (e.g. $x^2$ , $xy$ , $y$ ).
A constant term may have no letters (e.g. $-2$ ).

LIKE TERMS have the SAME letters with the SAME powers. Coefficient differences are fine.

$3x$ and $7x$ → like.
$5xy$ and $-2xy$ → like.
$4x^2$ and $4x$ → NOT like ( $x^2 \ne x$ ).
$3xy$ and $3yx$ → like ( $xy = yx$ ).

You can ADD or SUBTRACT like terms by combining their coefficients:

$5x + 3x = 8x.$ $7x^2y - 4x^2y = 3x^2y.$

You CANNOT combine UNLIKE terms. $3x + 5y$ stays as $3x + 5y$ .

Worked example. Simplify $4a + 7b - 2a + 5b - 3$ .

Group like: $(4a - 2a) + (7b + 5b) - 3$ .
Combine: $2a + 12b - 3$ .

Like terms: same letters, same powers.
Add coefficients of like terms.
Unlike terms stay separate.

Expanding brackets

Distribute the multiplication.

Single bracket: multiply everything inside by what's outside.

$a(b + c) = ab + ac.$

Worked example. Expand $3(2x - 5y + 4)$ .

$3 \times 2x = 6x$ .
$3 \times (-5y) = -15y$ .
$3 \times 4 = 12$ .
Result: $6x - 15y + 12$ .

Two brackets (FOIL). $(a + b)(c + d) = ac + ad + bc + bd$ .

The acronym FOIL (First, Outer, Inner, Last) helps you remember to multiply every term in the first bracket by every term in the second.

Worked example. Expand $(x + 3)(x + 5)$ .

F: $x \times x = x^2$ .
O: $x \times 5 = 5x$ .
I: $3 \times x = 3x$ .
L: $3 \times 5 = 15$ .
Sum: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$ .

Worked example (Extended). Expand $(2x - 3)(x + 4)$ .

F: $2x \times x = 2x^2$ .
O: $2x \times 4 = 8x$ .
I: $-3 \times x = -3x$ .
L: $-3 \times 4 = -12$ .
Sum: $2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$ .

Three special identities to memorise:

$(a+b)^2 = a^2 + 2ab + b^2$ .
$(a-b)^2 = a^2 - 2ab + b^2$ .
$(a+b)(a-b) = a^2 - b^2$ (difference of squares).

These show up CONSTANTLY in factorisation and algebra problems.

Single bracket: distribute the multiplier.
Double bracket: FOIL — every term from one × every term from the other.
$(a+b)^2 = a^2 + 2ab + b^2$ .
$(a-b)(a+b) = a^2 - b^2$ .

Substitution and evaluating

Replace letters with values — use brackets.

Substitution means replacing each letter with its given value, then evaluating.

Worked example. Find the value of $3x^2 - 2x + 5$ when $x = -4$ .

ALWAYS use brackets around the substituted value:

$3(-4)^2 - 2(-4) + 5$ .
BIDMAS: $(-4)^2 = 16$ .
$3 \times 16 - 2 \times (-4) + 5 = 48 + 8 + 5 = 61$ .

(Without brackets, $-4^2$ would be $-16$ — wrong by sign.)

Worked example. Find the value of $\dfrac{a^2 - b^2}{2a + b}$ when $a = 5$ and $b = 3$ .

Top: $5^2 - 3^2 = 25 - 9 = 16$ .
Bottom: $2(5) + 3 = 13$ .
Result: $\dfrac{16}{13}$ .

Multi-step. Sometimes you need to substitute into one formula then into another:

Volume of a cone: $V = \tfrac{1}{3}\pi r^2 h$ . Find $V$ when $r = 3$ and $h = 5$ .

$V = \tfrac{1}{3} \times \pi \times 3^2 \times 5 = \tfrac{1}{3} \times \pi \times 9 \times 5 = 15\pi \approx 47.1$ .

A simple rule: at the start of any substitution problem, REWRITE the expression with brackets around each substituted value. This single habit removes most sign errors.

Always BRACKET substituted values.
Apply BIDMAS.
Negative values especially need brackets.

Quick recap

Like terms: same letters, same powers. Add coefficients.
Single bracket: distribute.
Double bracket: FOIL.
$(a+b)^2 = a^2 + 2ab + b^2$ . $(a-b)(a+b) = a^2 - b^2$ .
Substitution: BRACKET each value.

Memorise this

Verbatim phrases and definitions MYP criterion-A markschemes credit.

$(a+b)^2 = a^2 + 2ab + b^2$
$(a-b)^2 = a^2 - 2ab + b^2$
$(a+b)(a-b) = a^2 - b^2$
$a(b+c) = ab + ac$
FOIL: First, Outer, Inner, Last

How it’s examined

Criterion A: simplify and expand. Criterion C: clear bracket-by-bracket working. Criterion D: substitute values into formulae from real contexts.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

Take this whole topic with you

Step-by-step worked examples — Algebraic Expressions

Step-by-step solutions to past-paper-style questions on algebraic expressions, written exactly the way a tutor would explain them at the board.

1Collect like terms
Getting started• simplify
Question
Simplify $5a + 3b - 2a + 7b - 4$ .
Step-by-step solution
1. Step 1
  Group like terms: $(5a - 2a) + (3b + 7b) - 4$ .
2. Step 2
  Combine: $3a + 10b - 4$ .
Answer
$3a + 10b - 4$
2Expand two brackets
Building confidence• expand
Question
Expand and simplify $(x - 4)(x + 7)$ .
Step-by-step solution
1. Step 1
  FOIL: First $x \times x = x^2$ . Outer $x \times 7 = 7x$ . Inner $-4 \times x = -4x$ . Last $-4 \times 7 = -28$ .
2. Step 2
  Sum: $x^2 + 7x - 4x - 28 = x^2 + 3x - 28$ .
Answer
$x^2 + 3x - 28$
3Use a special identity
Building confidence• identity
Question
Expand $(3x - 2)^2$ without using FOIL.
Step-by-step solution
1. Step 1
  Use $(a-b)^2 = a^2 - 2ab + b^2$ with $a = 3x$ and $b = 2$ .
2. Step 2
  $a^2 = 9x^2$ , $2ab = 2 \times 3x \times 2 = 12x$ , $b^2 = 4$ .
3. Step 3
  Result: $9x^2 - 12x + 4$ .
Answer
$9x^2 - 12x + 4$
4Substitute a negative
Stretch• substitution
Question
Find the value of $2x^2 - 5x + 3$ when $x = -3$ .
Step-by-step solution
1. Step 1
  Substitute WITH brackets: $2(-3)^2 - 5(-3) + 3$ .
2. Step 2
  BIDMAS: $(-3)^2 = 9$ . Expression: $2(9) - 5(-3) + 3$ .
3. Step 3
  Multiplications: $18 + 15 + 3$ .
4. Step 4
  Sum: $36$ .
Answer
$36$

Key Definitions and Keywords — Algebraic Expressions

Definitions to memorise and the exact keywords mark schemes credit for algebraic expressions answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Variable
Examiner keyword
A letter standing for an unknown or generic number.
Term
Examiner keyword
A single number, a single letter, or a product of these.
Coefficient
Examiner keyword
The numerical part of a term. The coefficient of $3x$ is 3.
Like terms
Examiner keyword
Terms with the same letters at the same powers. Can be combined by adding coefficients.
Expand
Remove brackets by multiplying out.

Common Mistakes and Misconceptions — Algebraic Expressions

The traps other students keep falling into on algebraic expressions questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Combining $3x^2$ with $3x$ .
Why it happens
Same letter — treated as like.
How to avoid it
Like terms need the same POWER too. $3x^2$ and $3x$ are NOT like; they cannot be combined.
✕Distributing $-(2x + 3) = -2x + 3$ .
Why it happens
Forgetting the negative applies to BOTH terms.
How to avoid it
$-(2x + 3) = -2x - 3$ . Distribute the minus to EVERY term.
✕Saying $(x + 3)^2 = x^2 + 9$ .
Why it happens
Squaring each term separately.
How to avoid it
$(x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9$ . You MUST expand with FOIL or use $(a+b)^2 = a^2 + 2ab + b^2$ .

Algebraic Expressions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Algebraic Expressions

Short Notes - Algebraic Expressions

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Collect like terms

2Expand two brackets

3Use a special identity

4Substitute a negative

Variable

Term

Coefficient

Like terms

Expand

✕Combining 3x23x^23x2 with 3x3x3x.

✕Distributing −(2x+3)=−2x+3-(2x + 3) = -2x + 3−(2x+3)=−2x+3.

✕Saying (x+3)2=x2+9(x + 3)^2 = x^2 + 9(x+3)2=x2+9.

Algebraic Expressions — frequently asked questions

Algebraic Expressions

Short Notes - Algebraic Expressions

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Collect like terms

2Expand two brackets

3Use a special identity

4Substitute a negative

Variable

Term

Coefficient

Like terms

Expand

✕Combining 3x23x^23x2 with 3x3x3x.

✕Distributing −(2x+3)=−2x+3-(2x + 3) = -2x + 3−(2x+3)=−2x+3.

✕Saying (x+3)2=x2+9(x + 3)^2 = x^2 + 9(x+3)2=x2+9.

Algebraic Expressions — frequently asked questions

✕Combining $3x^2$ with $3x$ .

✕Distributing $-(2x + 3) = -2x + 3$ .

✕Saying $(x + 3)^2 = x^2 + 9$ .

✕Combining $3x^2$ with $3x$ .

✕Distributing $-(2x + 3) = -2x + 3$ .

✕Saying $(x + 3)^2 = x^2 + 9$ .