1. Algebraic Fractions

2. Algebraic Fractions – Core 3, Chapter 1
Session 1
LO: Add, subtract, multiply and divide algebraic fractions.
Session 2
LO: Write improper algebraic fractions as mixed numbers using division or the remainder theorem.

3. Work out the answers to the following:
2 5 − 1 3
2 5 × 1 3
2 5 ÷ 1 3

4. Simplifying algebraic fractions
Cancel down by finding common factors in the numerator and the denominator.
Factors must be common to ALL TERMS (no cancelling randomly through addition signs).
Remove fractions from numerators and denominators by multiplying through by the appropriate number.
If you see a quadratic, a good tip is to factorise that first. One of its pair of brackets will usually cancel with another.
Difference of two squares: (x – y)² = x² – y².

5. Multiplying and dividing algebraic fractions
To multiply fractions, multiply the tops and multiply the bottoms.
To divide, flip the second fraction upside down and change to a multiply (you’re using the reciprocal here).
Cancel any common factors first – this will make it much easier.
Think about factorising quadratics and again, difference of two squares!

6. Adding and subtracting algebraic fractions
To add and subtract fractions, the denominators must be the same.
You need to multiply each fraction (numerator and denominator) by a number or expression to get a common denominator.
Write it out in full – don’t try to skip steps until you’re feeling really confident!
Expand brackets in the numerator and simplify where possible.
And again, FACTORISE QUADRATICS AND DIFFERENCE OF TWO SQUARES!

7. Writing in ‘mixed’ number form
Either use long division or the remainder theorem. You need to be able to do both methods.
Long division is just like in Core 2, but you will have a remainder. This is always divided by the original denominator.
Using the Remainder theorem:
F(X) = (Ax² + Bx + C)(divisor) + D
Substitute the value of x to make divisor = 0.
Substitute x = 0 to give an equation in C and D.
Equate coefficients in x³ and x².
The remainder will always be of order one less than the divisor (e.g. if the divisor is quadratic, the remainder will be linear).