Factorisation of Linear Algebraic Expressions

TL;DR

Factorisation means writing an expression as a product of its factors. For linear expressions, you look for the greatest common factor (GCF) of all terms. You then "undo" the distributive property to pull that GCF out front.

1. The Mental Model

Think of factorisation like un-packaging. You're starting with a sum of terms and trying to find a common "ingredient" you can pull out, leaving the rest of the ingredients inside a new package (parentheses).

2. The Core Material

Factorisation of linear algebraic expressions is the opposite of expanding. When you expand, you multiply a term by everything inside parentheses (e.g., $2(x+3) = 2x + 6$). When you factorise, you're starting with something like $2x+6$ and trying to get back to $2(x+3)$.

The main idea is to find what's common to all terms in the expression. This "common thing" is called the greatest common factor (GCF).

How to Find the GCF

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1. Look at the numbers (coefficients): Find the largest number that divides into all the coefficients evenly.
2. Look at the variables: See which variables are present in all terms. If a variable is common, take it with the lowest power it appears in any term. For linear expressions, this will usually just be the variable itself (e.g., $x$, not $x^2$).

Let's break down the process:

graph TD
    A["Start with expression (e.g., 6x + 9)"] --> B["Identify all terms (6x, 9)"];
    B --> C["Find GCF of numerical coefficients (GCF of 6, 9 is 3)"];
    C --> D{"Are there common variables in ALL terms?"};
    D -- "No" --> E["GCF is just the number (3)"];
    D -- "Yes (e.g., in 6x + 3x)" --> F["GCF includes variable with lowest power (3x)"];
    E --> G["Divide each original term by the GCF"];
    F --> G;
    G --> H["Write GCF outside parentheses"];
    H --> I["Write results of division inside parentheses"];
    I --> J["Final factorised expression (e.g., 3(2x + 3))"];

Applying the GCF

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Once you've found the GCF:

1. Write the GCF outside a set of parentheses.
2. Divide each term in the original expression by the GCF.
3. Write the results of these divisions inside the parentheses.

Remember, you can always check your answer by expanding your factorised expression to see if you get back to the original.

3. Worked Example

Let's factorise the expression $12a - 18$.

1. Identify terms: The terms are $12a$ and $-18$.
2. Find the GCF of the numerical coefficients: The coefficients are $12$ and $18$.
   • Factors of $12$: $1, 2, 3, 4, 6, 12$
   • Factors of $18$: $1, 2, 3, 6, 9, 18$
     The greatest common factor of $12$ and $18$ is $6$.
3. Find common variables: The term $12a$ has the variable $a$, but the term $-18$ doesn't. So, there are no common variables.
4. Determine the overall GCF: Since there are no common variables, the GCF is just $6$.
5. Divide each original term by the GCF:
   • $12a \div 6 = 2a$
   • $-18 \div 6 = -3$
6. Write the factorised expression: Put the GCF outside and the results of the division inside parentheses.
   $6(2a - 3)$

To check: $6 \times 2a = 12a$ and $6 \times -3 = -18$. So, $6(2a - 3) = 12a - 18$. It matches!

4. Key Takeaways

• Factorisation is the reverse of expanding an expression.
• You're looking for the greatest common factor (GCF) shared by all terms.
• The GCF includes the largest common number and any common variables (with their lowest power).
• Once you find the GCF, you divide each term by it and place the GCF outside parentheses.
• Always check your work by expanding your factorised expression back to the original.
• The sign of the terms inside the parentheses matters; be careful when dividing by a positive or negative GCF.

Common mistakes to avoid:
- Not finding the greatest common factor (e.g., factoring $4x+8$ as $2(2x+4)$ instead of $4(x+2)$).
- Forgetting to divide all terms by the GCF.
- Making sign errors when dividing negative terms.
- Incorrectly handling variables (e.g., thinking $x$ is common to $2x+3$ when it's not in the '3').

5. Now Try It

Factorise the following expressions:
a) $15y + 25$
b) $9m - 12n$
c) $4x - 10y + 2$

For each, find the GCF first, then show the division of each term, and finally write the fully factorised expression. Success looks like you getting the correct GCF and the correct terms inside the parentheses for all three.