To factorise in algebra, you're essentially reversing the process of expanding brackets. The goal is to rewrite an expression as a product of its factors. The most common method involves finding the highest common factor (HCF).
Here's a breakdown of how to factorise expressions:
1. Identify the Highest Common Factor (HCF):
- Look for the largest number that divides evenly into all the terms in the expression.
- Identify any variables that are common to all the terms, and find the lowest power of each common variable.
- The HCF is the product of these common numerical and variable factors.
Example: Factorise 6x + 9.
- The HCF of 6 and 9 is 3.
xis only in the first term, so it's not a common factor.- Therefore, the HCF of
6x + 9is 3.
2. Divide Each Term by the HCF:
- Divide each term in the original expression by the HCF you found in step 1.
Example (continued):
6x ÷ 3 = 2x9 ÷ 3 = 3
3. Write the Factored Expression:
- Write the HCF outside a set of brackets.
- Inside the brackets, write the result of dividing each term by the HCF.
Example (continued):
- The factored expression is
3(2x + 3).
4. Verification (Optional but Recommended):
- Expand the factored expression to check if it matches the original expression.
3(2x + 3) = 3 * 2x + 3 * 3 = 6x + 9. This confirms the factorisation is correct.
More Complex Examples and Techniques:
-
Factorising Quadratics: Expressions like
x² + 5x + 6require different techniques, often involving finding two numbers that add up to the coefficient of thexterm and multiply to give the constant term. -
Difference of Two Squares: Expressions in the form
a² - b²can be factored as(a + b)(a - b). For example,x² - 4 = (x + 2)(x - 2). -
Factorising by Grouping: Used for expressions with four or more terms, where you group terms together and factorise each group separately before finding a common bracketed term.
In summary, factorising in algebra involves identifying common factors and rewriting an expression as a product.
