Multiplying and Dividing Algebraic Fractions
You should have prior knowledge of factorising simple expressions and quadratics. Basic knowledge of fractions will be useful here.
Multiplying fractions
In this section we shall look at multiplying fractions through examples;
It’s very easy when multiplying fractions. You simply multiply numerators together and then multiply the denominators together as well. For example;
We can continue further and simplify by cancelling. We divide both 6 and 120 by 6.
Dividing 6 by 6 leaves 1 and 120 by 6 leaves 20.
We can also simplify first before working out the multiplication. Here is the fraction multiplication that we have to work out again.
We can cancel out the fractions diagonally and vertically before we multiply. In the fraction we can see that 2 cancels with 8.
…and 3 cancels with 15.
After cancelling out we can simply multiply the remaining numerators and denominators.
The last step offers much easier numbers to work with since it’s much easier to cancel small numbers such as 4 and 2 than larger numbers.
Algebraic Fractions multiplication
In this section we shall explore examples involving algebraic fractions multiplication. The same steps apply here as well.
And then cancel out the 8 and 4.
This should provode the following result;
Letters can also be cancelled out. The letters ms can be cancelled out.
…and the ns can also be cancelled out.
Lastly multiply what is left together.
A difficult alternative would have been to multiply first then cancel lastly.
Algebraic fractions division
Division is closely similar to multiplication expect we have to carry out one important step first. In this section we shall look at algebraic fractions division.
The important step is to turn the second fraction upside down and change the division sign to multiplication as shown below.
Now we can simply carryout the multiplication steps as we did above; First we cancel out 3 and 9.
Then we cancel out the 4 and 2.
We can also cancel out the letters. We can see that the as can cancel out each other.
…as well as the bs
We can now simply just multiply the remaining together;
Cancelling linear factors
In this section we shall look at a much complex examples than what we have seen above. We shall explore examples involving cancelling linear factors. The steps are not different to what we have seen so far.
First we look for any numbers that can cancel out. We can see that none can since 3 cannot cancel with either 2 and 4. The variables xs can cancel out. We can see that x can cancel with x3.
The linear factors are obvious. We can see that (x+7) can be cancelled from the top and bottom.
Lastly we multiply what is left as we did above.
This results in the following fraction;
Algebraic Fractions division
In this section we shall look at fractions involving division.
We have to work out the following fraction.
First let’s look at the common factors and put the expressions in brackets. we can see that 4x+12 has a common factor of 4. So this could become
and 6x2-12x-90 has a common factor of 6 so this could become
5x-10 has a common factor of 5 so that could become
and x2-4 has no common factors. The result is;
The next step would be to factorise the quadratics. X2-2x-15 is the same as (x+3)(x-5) and x2 – 4 is the same as (x+2)(x-2).
The second fraction can now be turned upside down and change the division sign to multiplication as we did before.
Cancel out the common factors. 4 and 6 can be divided by 2.
We can also cancel out the factor (x+3)
Continue cancelling out more factors. The factor (x-2) can be cancelled out.
Find the final answer by multiplying everything else together.
