MYTUTOR SUBJECT ANSWERS

276 views

How would you express (11+x-x^2)/[(x+1)(x-2)^2] in terms of partial fractions?

The first thing to consider when converting an expression to impartial fractions is how many fractions you will have in the end. With linear (no powers) brackets, found in the denominator of your initial fraction, the number of partial fractions at the end of your workings coincides with the number of said brackets. For example, a fraction with denominator (x+5)(x-2) would mean a final answer consisting of 2 fractions. However, in cases involving powers, such as this one, there will be an additional fraction in the answer. So, in this case, (x+1)(x-2)^2 will mean an answer consisting of 3 brackets.

            Next, we write our fraction out into a kind of placeholder, with A, B and C used in our numerators to represent the answers we hope to calculate. The denominators will be our original brackets on the denominator of the question, each separated into their own fraction, as follows:

(11+x-x^2)/[(x+1)(x-2)^2] = A/(x+1) + B/(x-2) + C/(x-2)^2

Notice that the B and C fractions are almost identical apart from the power of 2 added to only one of the brackets. This power, as well as the linear form in B, ensures we have partial fractions that, if added together once again, would create the original expression in the question.

Next, we multiply through by the original denominator, the brackets. This will create the equation [1] as follows:

[1]   11+x-x^2 = A(x-2)^2 + B(x+1)(x-2) + C(x+1)

An easy way to work out what your expression should look like in this step is to compare what you have in your brackets through which you are multiplying, and then to look at what is in the denominator of each fraction. Each term (A,B,C etc) will essentially be multiplied by what it does not have in its denominator, as the bracket it does have will cancel. We will use A in the long-hand way as an example:

[A/(x+1)] * [(x+1)(x-2)^2] = [A(x+1)(x-2)^2)]/(x+1) = A(x-2)^2

In this example, the two (x+1) brackets, top and bottom, cancelled, leaving us with the answer. The next step is to look at the brackets paired with our letters, and determine an x value that will equal as many terms as possible to zero. Lets take x=-1 , and substitute this into equation [1]. This will mean our B and C fractions will equal to zero, as follows:

11+(-1)-(-1)^2 = A(-1-2)^2 + B(0) + C(0) = A(9)

We can now rearrange this to find A:

9 = 9A

A = 1

We then repeat this method with new x values to find the value of C:

x=2

11+2-4 = A(0) + B(0) + C(3)

3C = 9

C = 3

To find B, we can substitute in our values for A and C, as well as any x value (other than those already used):

x=1

11+1-1 = 1(-1)^2 + B(2)(-1) + 3(2)

-2B = 11-1-6

B = 4/-2 = -2

To complete our answer, we rewrite equation [1], with our values for A, B and C substituted in:

(11+x-x^2)/[(x+1)(x-2)^2] = 1/(x+1) – 2/(x-2) + 3/(x-2)^2

This is our answer, in partial fractions.

Matthew M. A Level Maths tutor, GCSE Maths tutor, GCSE History tutor,...

7 months ago

Answered by Matthew, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

266 SUBJECT SPECIALISTS

£36 /hr

James G.

Degree: Mathematical Physics (Doctorate) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.STEP.

“Currently a 3rd year PhD student in Mathematical Physics. I'm very passionate about teaching as well as my subject area. Look forward to hearing from you.”

MyTutor guarantee

£20 /hr

Elliot D.

Degree: Mathematics (Bachelors) - St. Andrews University

Subjects offered:Maths, Physics

Maths
Physics

“Hi, I'm a 2nd year Computer Science student at St Andrews University. I mainly teach Maths at both GCSE and A-level.”

£20 /hr

Chris B.

Degree: Economics (Bachelors) - Durham University

Subjects offered:Maths, Further Mathematics + 2 more

Maths
Further Mathematics
Economics
-Personal Statements-

“About me: I am currently in my first year of studying Economics at Durham University. I have been tutoring maths for about two years now, so have had plenty of experience passing on my knowledge and interest in my subjects to younger ...”

About the author

£20 /hr

Matthew M.

Degree: Engineering Mathematics (Bachelors) - Bristol University

Subjects offered:Maths, Law+ 1 more

Maths
Law
History

“I am an Engineering Math student at the University of Bristol. I have a range of interests, which cover maths, history and law, and would love to help others in these areas as well as encourage the same enthusiasm I have in others! I a...”

MyTutor guarantee

You may also like...

Posts by Matthew

How effective is the Human Rights Act 1998 at protecting our human rights?

How far does the law in Engalnd and Wales protect us against indirect discrimination?

How would you express (11+x-x^2)/[(x+1)(x-2)^2] in terms of partial fractions?

Other A Level Maths questions

give the coordinates of the stationary points of the curve y = x^4 - 4x^3 + 27 and state with reason if they are minumum, maximum, or points of inflection.

Find the set of values of x for which 3x^2+8x-3<0.

How can I remember trig identities?

How to find the stationary point of y= x^2-108x^(1/2)+16 and determine the nature of the stationary point?

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok