1. Simplify algebraic fractions into the lowest terms
Objective
Simplify algebraic fractions into the lowest terms
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2. 12 15 What do we mean by ‘write in the lowest terms’ or ‘simplify’?
You can see that 12/15 is equal to 4/5 Here is how to do this with numbers

3. What do we mean by ‘write in the lowest terms’ or ‘simplify’?12 15 
4 x 3
5 x 3
Cancel out common factors
Factorise top and bottom

4. The remaining numbers give the fraction in its lowest terms
What do we mean by ‘write in the lowest terms’ or ‘simplify’? 12 
4 x 3 15
5 x 3  4 5
The remaining numbers give the fraction in its lowest terms

5. We can use the same method to simplify algebraic fractions into the lowest terms.x2
2x3 
x x x
2 x x x x x x  1 2x
This gives…
Factorise…
Cancel common terms…

6. Simplify these algebraic fractions y 4
2a2 a3  2 a 8h
4h2  2 h
3t3 6t  t2 2

7. x2 + 3x – 10 5 x -2 = -10 5 + -2 = 3 (x + 5)(x – 2) x2 + 3x x(x + 3)
Sometimes, the fractions will have quadratics as the numerator and/or denominator. You will need to factorise these and to refresh your memory, below are examples of how three different types of quadratic are factorised.
x2 + 3x – 10
5 x -2 = -10
= 3
(x + 5)(x – 2)
x2 + 3x
x(x + 3)
x2 - 16
(x + 4)(x – 4)
Common factor of x
Difference between two squares
Finding two numbers that have a product of -10 and sum of +3

8. x2 + 2x x2 - 4 x(x + 2) (x + 2)(x - 2) x (x – 2)  
Here is an example of how to simplify the algebraic fraction below:
Cancel common terms
The top terms have a common factor of x
x2 + 2x
x2 - 4
x(x + 2)
(x + 2)(x - 2) x
(x – 2)  
The bottom is the difference between two squares
This is the fraction in its lowest terms

9. x2 - 9 x2 + x - 6 (x + 3)(x – 3) (x + 3)(x - 2) x - 3 x - 2  
Another example of how to simplify a slightly more difficult the algebraic fraction:
The top is the difference between two squares
Cancel common terms
x2 - 9
x2 + x - 6
(x + 3)(x – 3)
(x + 3)(x - 2)
x - 3
x - 2  
Factorise the bottom by finding two numbers that have a product of -6 and sum of 1
This give the fraction in its lowest terms

10. Write these algebraic fraction in the simplest terms:1.
x2 + 4x
x2 - 16 2.
x2 - 9
x2 – 3x 3.
x2 + 5x + 6
x2 + 3x 4.
x2 + 5x
x2 - 25 5.
x2 - 16
x2 – 3x - 4 6.
x2 - x - 12
x2 – 9 7.
x2 + 7x + 10
x2 - 2x - 8 8.
x2 + x - 6
x2 + 8x + 15 9.
x2 - 5x - 6
x2 – 4x - 12

11. Here are the answers: 1. x
x - 4 2.
x + 3 x 3.
x + 2 x 4. x
x - 5 5.
x + 4
x + 1 6.
x - 4
x – 3 7.
x + 5
x - 4 8.
x - 2
x + 5 9.
x + 1
x + 2