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Sorting the Algebra Toolkit

Imagine emptying a pencil case onto a desk and grouping the pens, pencils, and highlighters together. In algebra, we do exactly the same thing to tidy up a mathematical expression. An expression is a phrase containing numbers, letters, and operators, but it does not have an equals sign. It is built out of individual blocks called a term. A term can be a single number, a letter, or a combination of both multiplied together.

The letter in a term is called a variable because it represents an unknown or changing value, such as $x$ or $y$ .
The number multiplying the variable is the coefficient. For example, in the term $-4y^3$ , the coefficient is .

There are two golden rules for reading terms. First, watch out for the "invisible 1": if a variable has no number in front of it (like $x$ ), the coefficient is $1$ , and if it says $-x$ , the coefficient is $-1$ . Second, a plus ( $+$ ) or minus ( $-$ ) sign always belongs to the term immediately following it. If you move a term around to rearrange an expression, its sign must travel with it.

Identifying Like Terms

To simplify an expression fully, you must group and combine similar blocks together. We call these like terms. Two terms are only "like" if they have the exact same variables raised to the exact same powers.

$4x^2$ and $9x^2$ are like terms.
$3x$ and $3x^2$ are like terms because the powers are different.

Because of the commutative property of multiplication, the order of the letters inside a single term does not matter ( $ab$ is exactly the same as $ba$ ). Therefore, $2ab$ and $5ba$ are like terms and can be collected together. Similarly, the order in which you write the final terms does not change the mathematical meaning, though alphabetical order is standard practice.

How to Collect Like Terms

To collect like terms, you add or subtract their numerical coefficients while leaving the variable parts completely unchanged. If the coefficients add up to zero, the term disappears entirely—we never write $0x$ in a final answer.

Simplify fully: $8p - 3q - 5p + 7$

Step 1: Identify like terms.

The like terms are $8p$ and $-5p$ .
$-3q$ is a distinct term.
$+7$ is a constant.

Step 2: Group the terms, keeping the operational signs attached to their specific terms.

$8p - 5p - 3q + 7$

Step 3: Combine the coefficients.

$(8 - 5)p - 3q + 7$
$3p - 3q + 7$

Final Answer:

$3p - 3q + 7$

Simplify fully: $6y^2 + 5y - 2y^2 - y - 4$

Step 1: Identify like terms.

The $y^2$ terms: $6y^2$ and $-2y^2$
The terms: and (remember the invisible 1, so this is )

Step 2: Group the terms.

$(6y^2 - 2y^2) + (5y - 1y) - 4$

Step 3: Combine the coefficients.

$4y^2 + 4y - 4$

Final Answer:

$4y^2 + 4y - 4$

Simplify fully: $12c - 2cd - 4c + 5dc$

Step 1: Identify like terms.

The $c$ terms: $12c$ and $-4c$
The $cd$ terms: $-2cd$ and $+5dc$ (since and are the exact same variables in a different order)

Step 2: Group the terms.

$(12c - 4c) + (-2cd + 5cd)$

Step 3: Combine the coefficients.

$8c + 3cd$

Final Answer:

$8c + 3cd$

Students often try to combine terms with different powers, such as adding $4x$ and $3x^2$ to get $7x^2$ — these are unlike terms and cannot be combined.

Watch out for order of operations with signs. In the expression $10a - 3a + 2a$ , a frequent error is subtracting the sum of $3a$ and $2 a$ to get . You must evaluate left to right: , then .

In OCR exams, drawing loops or boxes around each term — making sure to include the plus or minus sign directly in front of it — will help you group terms correctly and can help secure the M1 method mark.

When an instruction says 'simplify fully', leaving an answer like $1x$ instead of $x$ is considered poor mathematical form and can occasionally lead to dropped accuracy marks.

A mathematical phrase containing terms, numbers, and operators, without an equals sign.

A single number, a variable, or numbers and variables multiplied together.

A letter or symbol representing a quantity that can change or is unknown.

The numerical multiplier of a variable in an algebraic term.

A term that consists only of a number and has no variable part.

Terms that contain the exact same variables raised to the exact same powers.

To write an expression in its most compact form by combining all like terms.

A mathematical phrase containing terms, numbers, and operators, without an equals sign.

A single number, a variable, or numbers and variables multiplied together.

A letter or symbol representing a quantity that can change or is unknown.

The numerical multiplier of a variable in an algebraic term.

A term that consists only of a number and has no variable part.

Terms that contain the exact same variables raised to the exact same powers.

To write an expression in its most compact form by combining all like terms.

Simplifying Expressions

Sorting the Algebra Toolkit

Identifying Like Terms

How to Collect Like Terms

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Key Terms(7)

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