To factorise fractions, you simplify them by identifying and canceling out common factors in the numerator and the denominator. The provided reference explains this method in detail.
Steps to Factorise Fractions
Here's a breakdown of how to factorise fractions, following the process mentioned in the reference:
- Factor the Numerator: Identify all the factors of the numerator. This involves finding numbers that multiply together to give the numerator.
- Factor the Denominator: Similarly, find all the factors of the denominator.
- Identify Common Factors: Compare the factors of the numerator and the denominator. Look for factors that appear in both.
- Cancel Common Factors: Divide both the numerator and the denominator by their common factors. This simplifies the fraction.
- Multiply Remaining Factors: After canceling, multiply the remaining factors in the numerator and the remaining factors in the denominator to get the simplified fraction.
Examples
Let's illustrate this process with some examples:
Example 1
Suppose we have the fraction 12/18:
- Numerator Factors: The factors of 12 are 1, 2, 3, 4, 6, and 12.
- Denominator Factors: The factors of 18 are 1, 2, 3, 6, 9, and 18.
- Common Factors: The common factors are 1, 2, 3, and 6.
- Cancel Common Factors: If we divide both by 6 (the greatest common factor), we get 12/6 = 2 and 18/6 = 3.
- Simplified Fraction: The simplified fraction is 2/3.
Example 2
Consider a more complex case with algebraic expressions: (x^2 + 2x) / (x^2 + 4x + 4)
- Factor Numerator: x^2 + 2x can be factored as x(x + 2)
- Factor Denominator: x^2 + 4x + 4 can be factored as (x + 2)(x + 2) or (x + 2)^2
- Identify Common Factors: The common factor is (x + 2)
- Cancel Common Factors: We cancel out one (x + 2) from both, leaving x in the numerator and (x + 2) in the denominator.
- Simplified Fraction: The simplified fraction is x / (x + 2).
Summary
| Step | Description |
|---|---|
| Factor Numerator | Find all the numbers that multiply to give the numerator |
| Factor Denominator | Find all the numbers that multiply to give the denominator |
| Common Factors | Identify the common numbers present in both numerator and denominator |
| Cancel Factors | Divide both the numerator and denominator by their common factors |
| Simplify | Multiply the remaining factors to obtain the reduced fraction |
By following these steps, you can effectively factorise fractions, making them simpler and easier to work with. Remember, the key is to identify the common factors and divide both numerator and denominator by them. The process, as explained in the reference, involves factoring both parts, canceling common elements, and multiplying the remaining factors.
