Linear Expressions: Simplifying and Single Brackets

Simplifying Expressions by Collecting Like Terms

Every time you sort your laundry, you group similar items together—socks with socks, shirts with shirts. In algebra, we do exactly the same thing by collecting like terms to make expressions easier to read.
An algebraic expression is a collection of terms made up of numbers, variables, and operation signs. A term is a single number, variable, or numbers and variables multiplied together.
Terms are only "like" if they contain exactly the same variable (letter) raised to the exact same powers. For example, $3x^2$ and $5x^2$ are like terms, but $3x$ and $3x^2$ are not. Remember that order does not matter due to the commutative property, so $xy$ and $yx$ can be collected together.
The coefficient is the numerical factor multiplying the variable. If a variable has no visible number, the coefficient is $1$ (so $x = 1x$ ), and a constant is a term with only a number and no variables.
The plus or minus sign in front of a term always belongs to that term and moves with it when reordering an expression.

Calculate: Simplify the expression $x^2 - 3xy + 2x^2 + 4x - 2xy - x + 7$

Step 1: Identify like terms (same variables and powers).
- $x^2$ terms: $+1x^2$ and $+2x^2$
- $xy$ terms: $-3xy$ and $-2xy$
- $x$ terms: $+4x$ and $-x$ (which is treated as $-1x$ )
- Constant terms: $+7$
Step 2: Group and calculate coefficients.
- $x^2$ group: $(1 + 2)x^2 = 3x^2$
- $xy$ group: $(-3 - 2)xy = -5xy$
- $x$ group: $(4 - 1)x = 3x$
- Constant group: $+7$
Step 3: Write the final simplified expression.
- Final answer: $3x^2 - 5xy + 3x + 7$

Multiplying a Single Term Over a Bracket

Why does this matter? Expanding brackets allows us to "unpack" complex algebraic structures so they can be simplified and solved.
Expanding (or multiplying out) uses the distributive law, meaning every term inside a single bracket must be multiplied by the factor directly outside.
Pay close attention to sign rules when dealing with negative terms: multiplying two of the same signs gives a positive answer ( $- \times - = +$ ), while different signs give a negative. Also remember your index laws, such as $x \times x = x^2$ and $x \times x^2 = x^3$ .

Calculate: Expand and simplify $10 - 2(3x - 4)$

Step 1: Expand the bracket. Multiply the factor outside ( $-2$ $- 2$ ) by every term inside.
- First term: $-2 \times 3x = -6x$
- Second term: $-2 \times -4 = +8$ (Negative multiplied by negative gives a positive)
Step 2: Substitute back into the expression.
- $10 - 6x + 8$
Step 3: Collect like terms.
- Final answer: $18 - 6x$

Taking Out Common Factors

You can easily mix sugar into tea, but you cannot easily separate it back out. In algebra, however, expanding brackets is completely reversible through a process called factorising.
To factorise fully, you must find the Highest Common Factor (HCF) of all terms in the expression. This includes both the numerical HCF (the largest integer divisor for the numbers) and the algebraic HCF (the lowest power of a variable present in all terms).
If a term in your expression is exactly identical to the HCF you are extracting, you must write a $1$ inside the bracket to hold its place.

Calculate: Factorise fully $12p^2q^3 + 9p^3q - 3pq$

Step 1: Identify the numerical HCF.
- The largest divisor that goes perfectly into $12$ , $9$ , and $-3$ is $3$ .
Step 2: Identify the algebraic HCF.
- Look for the lowest power of each variable present in every term. For $p$ , it is $p$ . For $q$ , it is $q$ .
Step 3: Combine to find the overall HCF.
- The overall HCF is $3pq$ . This goes outside the bracket.
Step 4: Divide each original term by the HCF to find the terms inside the bracket.
- First term: $12p^2q^3 \div 3pq = 4pq^2$
- Second term: $9p^3q \div 3pq = 3p^2$
- Third term: $-3pq \div 3pq = -1$ (Applying the "1" rule)
Step 5: Write the final factorised expression.
- Final answer: $3pq(4pq^2 + 3p^2 - 1)$

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Exam Tips

Common Mistake
Students often incorrectly combine unlike terms by adding everything together (e.g., $3x + 7 = 10x$ ); remember that terms with different letters or powers cannot be added or subtracted.
Common Mistake
When a negative number is outside a bracket, students often forget to change the sign of the second term inside (e.g., incorrectly expanding $-2(x - 5)$ as $-2x - 10$ instead of $-2x + 10$ ).
3
In AQA questions that say 'Factorise fully', extracting only a partial factor (like taking out just the numbers but leaving variables behind) will usually only earn 1 out of 2 marks.
4
If terms cancel out completely when collecting like terms (e.g., $5x - 5x$ ), do not write $0x$ in your final answer; the term simply disappears from the expression.
5
You can quickly check if you have factorised correctly by expanding your final bracketed answer; it must perfectly match the original expression.

Key Terms(16)

Algebraic expression: A collection of terms made up of numbers, variables, and operation signs.
Term: A single number, variable, or numbers and variables multiplied together, separated by + or - signs in an expression.
Like terms: Terms that contain exactly the same variables (letters) raised to the exact same powers.
Variable: A letter representing an unknown value or a quantity that can change.
Commutative property: The mathematical property stating that the order of values does not matter when adding or multiplying (e.g., $xy = yx$ ).
Coefficient: The numerical factor of a term (the number multiplying the variable).
Constant: A term that consists of only a number with no variables attached.
Expanding: The process of removing brackets from an expression by multiplying the factor outside by every term inside.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
Single bracket: An algebraic expression enclosed within one pair of brackets that is multiplied by a single outside term.
Factor: The term (number or variable) outside the bracket that acts as the multiplier.
Negative terms: Algebraic terms that are preceded by a minus sign, representing a value less than zero.
Index laws: The mathematical rules used for simplifying expressions involving powers (indices).
Factorise: To rewrite an expression as a product of its factors by introducing brackets.
Highest Common Factor (HCF): The largest numerical divisor or highest algebraic power that divides exactly into all given terms.
Divisor: A number or term by which another number or term is divided.

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Key Terms(16)

Algebraic expression: A collection of terms made up of numbers, variables, and operation signs.
Term: A single number, variable, or numbers and variables multiplied together, separated by + or - signs in an expression.
Like terms: Terms that contain exactly the same variables (letters) raised to the exact same powers.
Variable: A letter representing an unknown value or a quantity that can change.
Commutative property: The mathematical property stating that the order of values does not matter when adding or multiplying (e.g., $xy = yx$ ).
Coefficient: The numerical factor of a term (the number multiplying the variable).
Constant: A term that consists of only a number with no variables attached.
Expanding: The process of removing brackets from an expression by multiplying the factor outside by every term inside.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
Single bracket: An algebraic expression enclosed within one pair of brackets that is multiplied by a single outside term.
Factor: The term (number or variable) outside the bracket that acts as the multiplier.
Negative terms: Algebraic terms that are preceded by a minus sign, representing a value less than zero.
Index laws: The mathematical rules used for simplifying expressions involving powers (indices).
Factorise: To rewrite an expression as a product of its factors by introducing brackets.
Highest Common Factor (HCF): The largest numerical divisor or highest algebraic power that divides exactly into all given terms.
Divisor: A number or term by which another number or term is divided.

Linear Expressions: Simplifying and Single Brackets

Simplifying Expressions by Collecting Like Terms

Every time you sort your laundry, you group similar items together—socks with socks, shirts with shirts. In algebra, we do exactly the same thing by collecting like terms to make expressions easier to read.
An algebraic expression is a collection of terms made up of numbers, variables, and operation signs. A term is a single number, variable, or numbers and variables multiplied together.
Terms are only "like" if they contain exactly the same variable (letter) raised to the exact same powers. For example, $3x^2$ and $5x^2$ are like terms, but $3x$ and $3x^2$ are not. Remember that order does not matter due to the commutative property, so $xy$ and $yx$ can be collected together.
The coefficient is the numerical factor multiplying the variable. If a variable has no visible number, the coefficient is $1$ (so $x = 1x$ ), and a constant is a term with only a number and no variables.
The plus or minus sign in front of a term always belongs to that term and moves with it when reordering an expression.

Calculate: Simplify the expression $x^2 - 3xy + 2x^2 + 4x - 2xy - x + 7$

Step 1: Identify like terms (same variables and powers).
- $x^2$ terms: $+1x^2$ and $+2x^2$
- $xy$ terms: $-3xy$ and $-2xy$
- $x$ terms: $+4x$ and $-x$ (which is treated as $-1x$ )
- Constant terms: $+7$
Step 2: Group and calculate coefficients.
- $x^2$ group: $(1 + 2)x^2 = 3x^2$
- $xy$ group: $(-3 - 2)xy = -5xy$
- $x$ group: $(4 - 1)x = 3x$
- Constant group: $+7$
Step 3: Write the final simplified expression.
- Final answer: $3x^2 - 5xy + 3x + 7$

Multiplying a Single Term Over a Bracket

Why does this matter? Expanding brackets allows us to "unpack" complex algebraic structures so they can be simplified and solved.
Expanding (or multiplying out) uses the distributive law, meaning every term inside a single bracket must be multiplied by the factor directly outside.
Pay close attention to sign rules when dealing with negative terms: multiplying two of the same signs gives a positive answer ( $- \times - = +$ ), while different signs give a negative. Also remember your index laws, such as $x \times x = x^2$ and $x \times x^2 = x^3$ .

Calculate: Expand and simplify $10 - 2(3x - 4)$

Step 1: Expand the bracket. Multiply the factor outside ( $-2$ $- 2$ ) by every term inside.
- First term: $-2 \times 3x = -6x$
- Second term: $-2 \times -4 = +8$ (Negative multiplied by negative gives a positive)
Step 2: Substitute back into the expression.
- $10 - 6x + 8$
Step 3: Collect like terms.
- Final answer: $18 - 6x$

Taking Out Common Factors

You can easily mix sugar into tea, but you cannot easily separate it back out. In algebra, however, expanding brackets is completely reversible through a process called factorising.
To factorise fully, you must find the Highest Common Factor (HCF) of all terms in the expression. This includes both the numerical HCF (the largest integer divisor for the numbers) and the algebraic HCF (the lowest power of a variable present in all terms).
If a term in your expression is exactly identical to the HCF you are extracting, you must write a $1$ inside the bracket to hold its place.

Calculate: Factorise fully $12p^2q^3 + 9p^3q - 3pq$

Step 1: Identify the numerical HCF.
- The largest divisor that goes perfectly into $12$ , $9$ , and $-3$ is $3$ .
Step 2: Identify the algebraic HCF.
- Look for the lowest power of each variable present in every term. For $p$ , it is $p$ . For $q$ , it is $q$ .
Step 3: Combine to find the overall HCF.
- The overall HCF is $3pq$ . This goes outside the bracket.
Step 4: Divide each original term by the HCF to find the terms inside the bracket.
- First term: $12p^2q^3 \div 3pq = 4pq^2$
- Second term: $9p^3q \div 3pq = 3p^2$
- Third term: $-3pq \div 3pq = -1$ (Applying the "1" rule)
Step 5: Write the final factorised expression.
- Final answer: $3pq(4pq^2 + 3p^2 - 1)$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often incorrectly combine unlike terms by adding everything together (e.g., $3x + 7 = 10x$ ); remember that terms with different letters or powers cannot be added or subtracted.
Common Mistake
When a negative number is outside a bracket, students often forget to change the sign of the second term inside (e.g., incorrectly expanding $-2(x - 5)$ as $-2x - 10$ instead of $-2x + 10$ ).
3
In AQA questions that say 'Factorise fully', extracting only a partial factor (like taking out just the numbers but leaving variables behind) will usually only earn 1 out of 2 marks.
4
If terms cancel out completely when collecting like terms (e.g., $5x - 5x$ ), do not write $0x$ in your final answer; the term simply disappears from the expression.
5
You can quickly check if you have factorised correctly by expanding your final bracketed answer; it must perfectly match the original expression.

Key Terms(16)

Algebraic expression: A collection of terms made up of numbers, variables, and operation signs.
Term: A single number, variable, or numbers and variables multiplied together, separated by + or - signs in an expression.
Like terms: Terms that contain exactly the same variables (letters) raised to the exact same powers.
Variable: A letter representing an unknown value or a quantity that can change.
Commutative property: The mathematical property stating that the order of values does not matter when adding or multiplying (e.g., $xy = yx$ ).
Coefficient: The numerical factor of a term (the number multiplying the variable).
Constant: A term that consists of only a number with no variables attached.
Expanding: The process of removing brackets from an expression by multiplying the factor outside by every term inside.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
Single bracket: An algebraic expression enclosed within one pair of brackets that is multiplied by a single outside term.
Factor: The term (number or variable) outside the bracket that acts as the multiplier.
Negative terms: Algebraic terms that are preceded by a minus sign, representing a value less than zero.
Index laws: The mathematical rules used for simplifying expressions involving powers (indices).
Factorise: To rewrite an expression as a product of its factors by introducing brackets.
Highest Common Factor (HCF): The largest numerical divisor or highest algebraic power that divides exactly into all given terms.
Divisor: A number or term by which another number or term is divided.

Previous Subtopic: Algebraic VocabularyAlgebraic Vocabulary Next NoteLaws of Indices

Back to Expression Manipulation

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(16)

Algebraic expression: A collection of terms made up of numbers, variables, and operation signs.
Term: A single number, variable, or numbers and variables multiplied together, separated by + or - signs in an expression.
Like terms: Terms that contain exactly the same variables (letters) raised to the exact same powers.
Variable: A letter representing an unknown value or a quantity that can change.
Commutative property: The mathematical property stating that the order of values does not matter when adding or multiplying (e.g., $xy = yx$ ).
Coefficient: The numerical factor of a term (the number multiplying the variable).
Constant: A term that consists of only a number with no variables attached.
Expanding: The process of removing brackets from an expression by multiplying the factor outside by every term inside.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
Single bracket: An algebraic expression enclosed within one pair of brackets that is multiplied by a single outside term.
Factor: The term (number or variable) outside the bracket that acts as the multiplier.
Negative terms: Algebraic terms that are preceded by a minus sign, representing a value less than zero.
Index laws: The mathematical rules used for simplifying expressions involving powers (indices).
Factorise: To rewrite an expression as a product of its factors by introducing brackets.
Highest Common Factor (HCF): The largest numerical divisor or highest algebraic power that divides exactly into all given terms.
Divisor: A number or term by which another number or term is divided.