A Level Maths: Success With Algebraic Division
Introduction
At school you will have been taught division. Some do a process called long division and there are others who do short division but essentially the process is the same.
As part of the A Level Maths Course you are required to divide algebraic expressions which sounds awful. You will be using your knowledge of indices to help you but there are two methods that you can use. Both have advantages and disadvantages which are discussed.
Polynomial Long Division
Long division of polynomials is exactly the same as long division of arithmetic numbers.
There are two other methods that can also be adopted, which are comparing coefficients and also the grid method.
It should be noted that if you divide a polynomial of order 2 (i.e. the highest power is 2 and this will then be a quadratic) by a linear expression then the result will be an expression of order 1 (i.e. highest power is 1).
Similarly if a polynomial of order 3 (i.e. the highest power is 3 and this will be a cubic) is divided by a linear expression then the result will be an expression of order 2 (i.e. the highest power will be 2 which is a quadratic).
Method 1: Comparing coefficients
Divide 2 x^3-3 x^2+x-6 \text { by } x-2
As mentioned the result will be a quadratic so the general result will be of the form a x^2+b x+c
\left(2 x^3-3 x^2+x-6\right) \div(x-2)=a x^2+b x+cMultiplying both sides by (x-2) will give: \left(2 x^3-3 x^2+x-6\right)=\left(a x^2+b x+c\right)(x-2)
Multiplying out the right hand side will give: 2 x^3-3 x^2+x-6=a x^3-2 a x^2+b x^2-2 b x+c x-2 c
And this can also be written as: :
2 x^3-3 x^2+x-6 \equiv a x^3+x^2(b-2 a)+x(c-2 b)-2 cYou will note that the equal sign has been replaced by an identity sign as this is true for all values of x.
Now it is possible to compare coefficients, which are the numbers in front of x^3, x^2 \text { and also } x
The relevant terms on the left are compared to the relevant terms on the right so that the values of a, b and c can be found as shown.
Comparing the coefficients of x^3:
2=aComparing the coefficients of x^2:
\begin{gathered} -3-b-2 a \\ -3+4=b \\ b=1 \end{gathered}Comparing the coefficients of x:
\begin{gathered} 1-c-2 b \\ 1+2=c \\ c=3 \end{gathered}
Method 2: Long Division
Divide 2 x^3-3 x^2+x-6 \text { by } x-2 by long division.
This type of question is set up using the “bus stop” method and it will be written as follows:
Always start with the first term that you are dividing by which in this case is x.
The 2 x^3 is divided by x to give 2 x^2. and this is then written on top of the bus stop. The 2 x^2 is then multiplied b x-2 to give 2 x^3-4 x^2 .
Now the 2 x^3-4 x^2 should be surrounded by curly brackets and it is being subtracted from 2 x^3-3 x^2. The purpose of the brackets is to help avoid any confusion with signs.
When the subtraction is completed, x^2, is what is left and the “x” from the original polynomial is brought down next to the x^2 to give x^2+x.
The process then repeats in that x^2 is divided by x which is x and this is written on top of the bus stop. Multiply the x by x-2 to give x^2-2 xwhich is then subtracted from x^2+x to give 3 x.
The “-6″ is then brought down next to the 3x to give 3x-6. The process repeats in that 3x is divided by x to give 3. Multiply x-2 by 3 to give 3x-6 and this is subtracted from the 3x-6 before it to give 0, which indicates that there is no remainder.
At first appearance both these methods seem to be very odd but you need to remember that this is brand new maths, and brand new of anything means that you are going to need to continue to keep practising these methods. Before continuing with the rest of the article you should make sure that you are happy with the methods outlined above.
Try the questions again and see if you are able to obtain the same solution on your own. If you think that you still need some help, you can invest in the services of a UK based online maths tutor who will be a fully qualified teacher. They will give you additional guidance and support to help you overcome any gaps in knowledge and understanding.
Method 3: The grid method
Divide 2 x^3-3 x^2+x-6 \text { by } x-2
First have what you are dividing by set up as shown:
The first term in the polynomial is 2 x^3 and so x needs to be multiplied by 2 x^2 . Multiplying 2 x^2 \text { by } x-2 \text { gives } 2 x^3 \text { and }-4 x^2 \text { as shown. }
Within the grid you want to have the terms that will give the original polynomial. -4 x^2 is given but -3 x^2 is needed, so x^2 needs to be added. In this process it is possible to add terms diagonally only.
Now the process can repeat. x needs to be multiplied by x \text { to give } x^2 \text { and } x multiplied by -2 gives -2x.
Now x is needed but within the grid -2x is shown so 3x needs to be added diagonally. From within the grid, to obtain 3x, x needs to be multiplied by 3 and multiplying 3 with -2 gives -6 which matches the last term of the polynomial and this shows that there is no remainder.
Using this method the answer is 2 x^2+x+3.
There is no better method than the other, it all depends on what you prefer and understand. Most textbooks tend to show the first two techniques. The grid method technique is rarely shown.
One of the best ways to revise A Level Maths is simply to do lots of questions. If you are not getting the correct answer however it is important that you do not ignore the question. You need to find out why you got something wrong. Quite often it can be a case of making a mistake with arithmetic but whatever the error is, do not ignore it.
To help boost revision many students attend one of our Easter A Level Maths Revision Courses which are classroom based and are focused to help you know the tricks that will help you get the best marks and the best grades in your final exams. Our Easter course is always the most popular as many schools and colleges ought to have finished teaching the content. These last few months are essential to consolidate your understanding and subject knowledge.
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