Algebraic Fractions

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.

Work out the answers to the following:

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.

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!

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!

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).