What is the primary difference between expanding and factorising?

Expanding removes brackets by multiplying a term across a sum, while factorising inserts brackets by dividing out the Highest Common Factor (HCF) to write the expression as a product.

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

Partial factorisation extracts any common factor, leaving potential factors still inside the bracket. Complete factorisation extracts the Highest Common Factor (HCF), leaving no common factors among the terms inside.

When factorising, how do you decide which power of a variable to take out as a factor?

You must take out the lowest power of the variable that appears in all terms. This ensures that the variable can be divided out of every term without resulting in negative exponents.

What is a common mistake when one of the terms in the expression is the same as the HCF?

Students often forget to write a '1' inside the bracket for that term. Because factorisation is division, a term divided by itself equals 1, which acts as a necessary placeholder.

What happens to the signs inside the bracket if you factorise out a negative term?

All the signs of the terms inside the bracket must be reversed. A positive term becomes negative, and a negative term becomes positive, because a negative times a negative equals a positive.

Why might a student get a factorisation like $2(4x + 6)$ marked as incomplete?

The student found a common factor (2), but not the Highest Common Factor (4). The terms inside the bracket (4x and 6) still share a common factor of 2, meaning it is not factorised fully.

Define the 'Highest Common Factor' (HCF) in the context of algebra.

The HCF is the product of the largest integer that divides all coefficients and the highest powers of variables that are common to every term in the expression.

What does it mean to 'factorise fully'?

It means to extract the greatest possible common factor from an expression so that the remaining terms inside the bracket have no common factors other than 1.

In the expression $7x^2y + 14xy^2$, what is the algebraic part of the HCF?

The algebraic HCF is $xy$. This is because $x$ is the lowest power of $x$ common to both, and $y$ is the lowest power of $y$ common to both.

Why is the number of terms inside the bracket always the same as the number of terms in the original expression?

Factorisation is a form of division that preserves the structure of the expression. Each original term must have a corresponding result after being divided by the common factor.

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Factorising Out Terms

Summary

Factorisation is the algebraic process of rewriting an expression as a product of its factors, effectively reversing the process of expanding brackets. By identifying the Highest Common Factor (HCF) of all terms, we can simplify expressions and prepare them for further operations like solving equations or simplifying fractions.

1. Definition & Core Concepts

Factorisation is the process of expressing an algebraic sum as a product of two or more factors. It is the mathematical inverse of expanding brackets, where a single term is multiplied across a sum.

A factor is a number or algebraic term that divides exactly into another term without leaving a remainder. In the expression $ab + ac$ , the term $a$ is a common factor of both $ab$ and $ac$ .

The goal of factorising out terms is to find the Highest Common Factor (HCF), which is the largest possible term that can be divided out of every part of the expression.

An area model showing a rectangle with height 'a' and width 'b+c'. The rectangle is split into two areas, 'ab' and 'ac', illustrating that a(b+c) = ab + ac.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Missing Terms: A common mistake is 'losing' a term when it is the same as the factor. If an expression has three terms, the factorised version must have three terms inside the brackets.
Incomplete HCF: Students often stop after finding a numerical factor but forget to look for algebraic factors (variables), or vice versa.
Sign Errors: When factorising expressions with subtraction, students often forget to carry the negative sign into the bracket or fail to change signs when factorising by a negative number.

Factorising Out Terms

Summary

1. Definition & Core Concepts

A factor is a number or algebraic term that divides exactly into another term without leaving a remainder. In the expression $ab + ac$ , the term $a$ is a common factor of both $ab$ and $ac$ .

The goal of factorising out terms is to find the Highest Common Factor (HCF), which is the largest possible term that can be divided out of every part of the expression.

An area model showing a rectangle with height 'a' and width 'b+c'. The rectangle is split into two areas, 'ab' and 'ac', illustrating that a(b+c) = ab + ac.

2. Underlying Principles

Factorisation relies on the Distributive Law of multiplication over addition, which states that $a(b + c) = ab + ac$ . Factorising simply applies this rule in the opposite direction.

The process is based on the principle of division. When we 'take out' a factor, we are essentially dividing every term in the expression by that factor and placing the result inside the brackets.

For an expression to be factorised, the factor placed outside must be a common divisor of every single term within the expression.

3. Methods & Techniques

Step 1: Find the Numerical HCF

Examine the coefficients (numbers) of each term and identify the largest integer that divides into all of them. For example, in $8x + 12$ , the HCF of $8$ and $12$ is $4$ .

Step 2: Find the Algebraic HCF

Look for variables that appear in every term. The HCF for a variable is the lowest power of that variable present in the expression. For instance, the HCF of $x^3$ and $x^2$ is $x^2$ .

Step 3: Combine and Extract

Multiply the numerical HCF and the algebraic HCF to get the total factor. Write this outside the brackets.

Step 4: Determine the Remaining Terms

Divide each original term by the HCF to find what remains. Write these results inside the brackets, maintaining the original signs ( $+$ or $-$ ).