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Question 11
Given that f(x) can be expressed in the form $$ \frac{A}{5x + 2} + \frac{B}{(5x + 2)^2} + \frac{C}{1 - 2x} $$ where A, B and C are constants. a) (i) find the value... show full transcript
Step 1
Answer
Substituting our known values into the form gives:
For expanding, usinging values derived:
After thorough substitution combined with filling from previous equations, it can be seen:
This results from systems of linear equations built from appending derived calculations and ultimately leads to the conclusion through direct substitution tests showing unwanted factors vanishing.
Step 2
Answer
Starting the expansion, we use binomial expansions on:
Combining these results would yield:
with each term corresponding to coefficients summarizing as constants p, q, and r surrounding conditions outlined for ranges to limit validity.
Step 3
Answer
The series for:\n( \frac{1}{5x+2} ) is valid where ( |5x + 2| > 0 ) which simplifies to: ( x > -\frac{2}{5} ).
For ( \frac{1}{1 - 2x} ), it requires the period where: ( |1 - 2x| > 0 ) which outlines similarly as: ( x < \frac{1}{2}. )
Hence, combining these results, the overall valid range for x is:
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