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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. |
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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 17 June 'Starter of the Day' page by Mr Hall, Light Hall School, Solihull: "Dear Transum, Comment recorded on the 17 November 'Starter of the Day' page by Amy Thay, Coventry: "Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils. |
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Featured ActivityPractice Papers Mathematics GCSE(9-1) Higher style questions and worked solutions presented as twenty short, free, practice papers to print out. |
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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. | ||
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❎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.
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❎
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."