Science:Math Exam Resources/Courses/MATH101/April 2007/Question 01 (f)

MATH101 April 2007

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Question 01 (f)
Write the form of the partial-fraction decomposition for ${\frac {10}{(x+1)^{2}(x^{2}+9)}}$ Do not determine the numerical values of these coefficients.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Remember that there are exceptions when you have repeated roots and quadratic factors. What were these exceptions?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First, we need to show that the factor $x^{2}+9$ is irreducible. Looking at the discriminant, we see that it is $b^{2}-4ac=0-4(1)(9)=-36<0$ and so this factor has no real roots. Thus, this is an irreducible quadratic. Hence, the partial fraction decomposition is ${\frac {10}{(x+1)^{2}(x^{2}+9)}}={\frac {A}{x+1}}+{\frac {B}{(x+1)^{2}}}+{\frac {Cx+D}{x^{2}+9}}$

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