Can every algebraic expression be factorised?

No. Some expressions have no common factors other than 1 across all terms. These are considered 'prime' or irreducible in the context of basic factorisation.

What does it mean for an expression to be written as a 'product of factors'?

It means the expression is written as two or more quantities multiplied together. In $3(x + 4)$, the factors are $3$ and $(x + 4)$.

In the expression $7xy + 14x$, what is the HCF?

The HCF is $7x$. The number $7$ is the largest factor of $7$ and $14$, and the variable $x$ appears in both terms, whereas $y$ only appears in the first.

How can you use expansion to verify a factorisation result?

Multiply the term outside the bracket by each term inside the bracket. If the resulting sum matches the original expression exactly, the factorisation is correct.

Why is the 'lowest power' rule used when factorising variables?

The lowest power represents the maximum amount of that variable that is common to all terms. For example, in $x^4 + x^2$, you can take $x^2$ out of both, but you cannot take $x^4$ out of $x^2$ without creating a fraction.

What is the fundamental difference between expanding and factorising?

Expanding is the process of multiplying out brackets to turn a product into a sum of terms. Factorising is the inverse process, where a sum of terms is turned into a product by identifying and extracting common factors.

How does 'partial factorisation' differ from 'full factorisation'?

Partial factorisation occurs when you extract a common factor, but not the largest one possible. Full factorisation requires extracting the Highest Common Factor (HCF) so that the terms remaining inside the bracket share no further commonalities.

When factorising $x^2 + x$, why is the answer $x(x + 1)$ and not just $x(x)$?

Because factorising is based on division. When you divide the second term ($x$) by the HCF ($x$), the result is $1$. If you omitted the $1$, expanding the bracket would only give you $x^2$, not the original $x^2 + x$.

What is a common mistake when identifying the HCF of variables with different powers, such as $a^3$ and $a^5$?

A common error is choosing the highest power ($a^5$) instead of the lowest power ($a^3$). You can only factorise out what is present in all terms; since $a^3$ is the largest amount contained within both, it is the HCF.

What should you do if the terms inside your bracket still share a common numerical factor after you have factorised?

This indicates that you have not found the Highest Common Factor. You must divide that remaining common factor out of the bracket and multiply it by the term already outside to achieve full factorisation.

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Factorising Out Terms

Summary

Factorising is the algebraic process of rewriting an expression as a product of its factors, effectively acting as the inverse operation to expanding brackets. By identifying and extracting the Highest Common Factor (HCF) from all terms in an expression, complex algebraic statements can be simplified into a more manageable form consisting of a multiplier outside a set of parentheses.

1. Definition & Core Concepts

Factorisation is the process of breaking down an algebraic expression into a product of simpler components called factors. In the context of 'factorising out terms', this specifically refers to identifying a common multiplier that exists across every term in the expression.

An expression is considered factorised when it is written as a term (the common factor) multiplied by a bracketed expression. For example, the expression $4x + 8$ is not factorised, whereas its equivalent form $4(x + 2)$ is factorised because it is expressed as the product of $4$ and $(x + 2)$ .

This process is the direct inverse of expanding brackets. While expansion removes parentheses by multiplying terms, factorisation introduces parentheses by dividing terms by their common factors.

Diagram showing factorising as the inverse of expanding brackets.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Factorising Out Terms

Summary

1. Definition & Core Concepts

This process is the direct inverse of expanding brackets. While expansion removes parentheses by multiplying terms, factorisation introduces parentheses by dividing terms by their common factors.

Diagram showing factorising as the inverse of expanding brackets.

2. Underlying Principles

The mathematical foundation of factorising is the Distributive Law, which states that $a(b + c) = ab + ac$ . Factorising simply applies this law in reverse to find the 'a' when given 'ab + ac'.

The process relies on the concept of the Highest Common Factor (HCF). To factorise an expression completely, one must find the largest possible number and the highest power of every variable that divides exactly into every term of the expression.

When a variable appears in multiple terms with different exponents, the HCF for that variable is always the lowest power present in those terms. For instance, in the expression $x^5 + x^3$ , the HCF is $x^3$ because it is the largest power that can be divided out of both terms without leaving a fractional exponent.

3. Methods & Techniques

Step-by-Step Factorisation

Step 1: Numerical HCF: Look at the coefficients (the numbers) of each term and determine the largest integer that divides into all of them. For $15x + 20$ , the numerical HCF is $5$ .
Step 2: Algebraic HCF: Look at the variables. If a variable appears in every term, identify the lowest power of that variable. For $x^3y^2 + x^2y^4$ , the algebraic HCF is $x^2y^2$ .
Step 3: Combine to find the overall HCF: Multiply the numerical and algebraic HCFs together to create the term that will sit outside the bracket.
Step 4: Determine the bracketed terms: Divide each original term by the overall HCF. The results of these divisions are placed inside the brackets, separated by the original operators ( $+$ or $-$ ).
Step 5: Final Assembly: Write the HCF followed by the bracketed terms, e.g., $HCF(Term1/HCF + Term2/HCF)$ .

4. Key Distinctions

It is vital to distinguish between partial factorisation and full factorisation. A partial factorisation extracts a common factor but leaves other common factors inside the bracket, whereas full factorisation extracts everything possible.

Feature	Partial Factorisation	Full Factorisation
Definition	Only some common factors are removed.	The Highest Common Factor (HCF) is removed.
Example	$2(4x + 6)$ for the expression $8x + 12$ .	$4(2x + 3)$ for the expression $8x + 12$ .
Result	Terms inside the bracket still share a factor.	Terms inside the bracket share no common factors (coprime).
Exam Status	Often awarded only partial marks.	Required for full marks.

5. Exam Strategy & Tips

The Expansion Check: Always verify your answer by mentally (or on scrap paper) expanding the brackets. If the result does not match the original expression exactly, an error occurred during division or HCF identification.
Watch the '1': A common mistake occurs when the HCF is identical to one of the terms in the expression. Remember that $Term \div HCF = 1$ , not $0$ . For example, factorising $5x + 5$ results in $5(x + 1)$ , not $5(x)$ .
Check the Brackets: After factorising, look at the terms inside the parentheses. If they still share a common factor (like both being even numbers), you have not factorised fully and must take out more.
Negative Factors: If the first term of an expression is negative, it is often cleaner to factorise out a negative HCF. This changes the signs of all terms inside the bracket.

6. Common Pitfalls & Misconceptions

Ignoring Variables: Students often focus only on the numbers and forget that variables (like $x$ or $y$ ) can also be part of the HCF if they appear in every term.
Incorrect Powers: A frequent error is taking out the highest power seen in the expression rather than the lowest power. You cannot divide $x^2$ by $x^3$ and remain with a polynomial term.
Sign Errors: When factorising expressions with subtraction, ensure the signs inside the bracket correctly reflect the original expression when multiplied back out.

The mathematical foundation of factorising is the Distributive Law, which states that $a(b + c) = ab + ac$ . Factorising simply applies this law in reverse to find the 'a' when given 'ab + ac'.

Step-by-Step Factorisation

Step 1: Numerical HCF: Look at the coefficients (the numbers) of each term and determine the largest integer that divides into all of them. For $15x + 20$ , the numerical HCF is $5$ .
Step 2: Algebraic HCF: Look at the variables. If a variable appears in every term, identify the lowest power of that variable. For $x^3y^2 + x^2y^4$ , the algebraic HCF is $x^2y^2$ .
Step 3: Combine to find the overall HCF: Multiply the numerical and algebraic HCFs together to create the term that will sit outside the bracket.
Step 4: Determine the bracketed terms: Divide each original term by the overall HCF. The results of these divisions are placed inside the brackets, separated by the original operators ( $+$ or $-$ ).
Step 5: Final Assembly: Write the HCF followed by the bracketed terms, e.g., $HCF(Term1/HCF + Term2/HCF)$ .

Feature	Partial Factorisation	Full Factorisation
Definition	Only some common factors are removed.	The Highest Common Factor (HCF) is removed.
Example	$2(4x + 6)$ for the expression $8x + 12$ .	$4(2x + 3)$ for the expression $8x + 12$ .
Result	Terms inside the bracket still share a factor.	Terms inside the bracket share no common factors (coprime).
Exam Status	Often awarded only partial marks.	Required for full marks.

The Expansion Check: Always verify your answer by mentally (or on scrap paper) expanding the brackets. If the result does not match the original expression exactly, an error occurred during division or HCF identification.
Watch the '1': A common mistake occurs when the HCF is identical to one of the terms in the expression. Remember that $Term \div HCF = 1$ , not $0$ . For example, factorising $5x + 5$ results in $5(x + 1)$ , not $5(x)$ .
Check the Brackets: After factorising, look at the terms inside the parentheses. If they still share a common factor (like both being even numbers), you have not factorised fully and must take out more.
Negative Factors: If the first term of an expression is negative, it is often cleaner to factorise out a negative HCF. This changes the signs of all terms inside the bracket.

Ignoring Variables: Students often focus only on the numbers and forget that variables (like $x$ or $y$ ) can also be part of the HCF if they appear in every term.
Incorrect Powers: A frequent error is taking out the highest power seen in the expression rather than the lowest power. You cannot divide $x^2$ by $x^3$ and remain with a polynomial term.
Sign Errors: When factorising expressions with subtraction, ensure the signs inside the bracket correctly reflect the original expression when multiplied back out.