This chapter covers multiplying and dividing algebraic fractions. By the end you should have a good understanding of how to multiply and divide algebraic fractions, understand how to factorise and cancel simple fractions first. You should have prior knowledge of factorising simple expressions and quadratics.
Lets look at multiplying fractions first. Suppose we wanted to multiply the following fractions below.
It’s very simple with multiplying fractions. You simply multiply numerators together and then multiply the denominators together as well.
We must now simplify by cancelling. We can divide both 3 6 and 120 by 6.
Dividing 6 by 6 leaves 1 and 120 by 6 leaves 20. There is also another method which could lead to the answer quicker. Here is our multiplication again.
We can cancel out the fractions diagonally and vertically before we multiply. Below we can see that 2 cancels with 8.
…and 3 cancels with 15.
After cancelling out we can simply multiply out the remaining as we did above.
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.
Below we’re going to work with algebraic fractions. The same steps above also apply here. Suppose we wanted to multiply the algebraic fractions below.
First we cancel out 3 and 15 as shown below.
And then cancel out the 8 and 4.
Then we re-write it as shown below.
We can also cancel out the letters. We can cancel out the ms.
…and the n as shown below.
And multiply what is left as shown below.
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. Suppose we wanted to divide the following fractions;
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/variables. 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
Below we shall look at a much complex of the above (cancelling linear factors) The steps are not different. Suppose we wanted to multiply the following;
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 through. We can see that x can cancel with x3.
Now we can observe the linear factors. Below we can see that (x+7) can be cancelled from the top and bottom.
Lastly we multiply what is left as we did above.
At this point best to leave the answers factorised as there is no need to multiply out the brackets.
Algebraic Fractions division
Here is another algebraic fraction division case and the same principle shown above applies. Let’s workout the division below;
First let’s look at the common factors and put the expressions in brackets. Above we can see that 4x+12 has a common factor of 4. So this could become 4(x+3) and 6x2-12x-90 has a common factor of 6 so this could become 6(x2-2x-12), 5x-10 has a common factor of 5 so that could become 5(x-2) 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).
Now we can turn the second fraction upside down and change the division sign to multiplication as we did above.
Then we can cancel out the common factors. 4 and 6 can be divided by 2.
We can also cancel out the factor (x+3)
Next we can cancel out the factor (x-2)
Then we multiply the remaining together.