|
|
Partial FractionsExercises on mastering the art of partial fraction decomposition. |
Express in partial fractions. Your answer must be in a particular format. Click the question number to see the template the answer must match.
InstructionsTry your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. |
||
|
|
||
|
More Activities: |
|
|
Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician? Comment recorded on the 1 August 'Starter of the Day' page by Peter Wright, St Joseph's College: "Love using the Starter of the Day activities to get the students into Maths mode at the beginning of a lesson. Lots of interesting discussions and questions have arisen out of the activities. Comment recorded on the 18 September 'Starter of the Day' page by Mrs. Peacock, Downe House School and Kennet School: "My year 8's absolutely loved the "Separated Twins" starter. I set it as an optional piece of work for my year 11's over a weekend and one girl came up with 3 independant solutions." |
Each month a newsletter is published containing details of the new additions to the Transum website and a new puzzle of the month. The newsletter is then duplicated as a podcast which is available on the major delivery networks. You can listen to the podcast while you are commuting, exercising or relaxing. Transum breaking news is available on Twitter @Transum and if that's not enough there is also a Transum Facebook page. |
|
Featured ActivityChristMaths Christmas activities make those December Maths lessons interesting, exciting and relevant. If students have access to computers there are some online activities to keep them engaged such as Christmas Ornaments and Christmas Light Up. |
AnswersThere are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer. A Transum subscription unlocks the answers to the online exercises, quizzes and puzzles. It also provides the teacher with access to quality external links on each of the Transum Topic pages and the facility to add to the collection themselves. Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members. If you would like to enjoy ad-free access to the thousands of Transum resources, receive our monthly newsletter, unlock the printable worksheets and see our Maths Lesson Finishers then sign up for a subscription now: Subscribe |
|
Go MathsLearning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school. Maths MapAre you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic. | ||
Teachers | ||
|
If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows: |
Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes. |
It may be worth remembering that if Transum.org should go offline for whatever reason, there is a mirror site at Transum.info that contains most of the resources that are available here on Transum.org. When planning to use technology in your lesson always have a plan B! |
|
Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments. |
||
© Transum Mathematics 1997-2024
Scan the QR code below to visit the online version of this activity.
https://www.Transum.org/go/?Num=1072
Close
❎Before starting this exercise you may want to check out Algebraic Fractions.
Refer to the syllabus for the course you are following to establish which levels are required.
Level 1 - Linear numerator and quadratic denominator decomposing to two fractions with linear denominators
Example:
$$ \frac{px+q}{(x-a)(x-b)} \equiv \frac{A}{x-a} + \frac{B}{x-b} $$Level 2 - Linear or quadratic numerator and cubic denominator decomposing to three fractions with linear denominators
Example:
$$ \frac{px^2+qx+r}{(x-a)(x-b)(x-c)} \equiv\frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} $$Level 3 - Linear or quadratic numerator and a denominator with a repeated factor
Examples:
$$ \frac{px+q}{(x-a)^2} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2} $$ $$ \frac{px^2+qx+r}{(x-a)^2(x-b)} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b} $$Level 4 - Linear or quadratic numerator and a denominator with a quadratic factor that cannot be factorised
Example:
$$ \frac{px^2+qx+r}{(x-a)(x^2+bx+c)} \equiv \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c} $$If the degree of the numerator is greater than or equal to the degree of the denominator you will need to divide first. See the exercise on Polynonial Division.
Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).
More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.
Express as partial fractions $$\frac{3x - 2}{x^2 + x - 12}$$
Solution:
We first factorise the denominator.
$$x^2 + x - 12$$ factorises as $$ (x+4)(x-3) $$
Thus, we rewrite the expression in the form:
$$\frac{3x - 2}{(x + 4)(x - 3)} = \frac{A}{x + 4} + \frac{B}{x - 3}$$
Where A and B are constants to be determined.
By multiplying through by the common denominator $$ (x+4)(x-3) $$
$$3x - 2 = A(x - 3) + B(x + 4)$$
Expanding and equating coefficients, we obtain:
$$3x - 2 = Ax - 3A + Bx + 4B$$
Combining like terms, this becomes:
$$3x - 2 = (A + B)x + (-3A + 4B)$$
Equating the coefficients of x and the constant terms, we have:
For the coefficients of x: $$3 = A + B$$
For the constant terms: $$-2 = -3A + 4B$$
Solving these equations simultaneously gives:
Let's substitute $$B = 3 - A$$ into the second equation:
$$-2 = -3A + 4(3 - A)$$
This simplifies to:
$$-2 = -3A + 12 - 4A$$
$$-2 = -7A + 12$$
$$-7A = -14$$
$$A = 2$$
Substituting $$A = 2$$ into $$B = 3 - A$$ gives:
$$B = 3 - 2 = 1$$
Therefore, the expression in partial fractions is:
$$\frac{3x - 2}{x^2 + x - 12} = \frac{2}{x + 4} + \frac{1}{x - 3}$$
These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.
Close
❎
Ann R,
Saturday, January 27, 2024
"Do you ever check your solutions on a graphics calculator? I plot the question and the answer and make sure that one graph sits on top of the other."