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The partial fraction decomposition is
The fraction is given as:
Split the fraction as follows:
Take the LCM
Cancel the common factors
8x + 19 = Ax - A + Bx + 8B
By comparison, we have:
Ax + Bx = 8x
-A + 8B = 19
This gives
A + B = 8
-A + 8B = 19
Add both equations
9B = 27
Divide by 9
B = 3
Substitute B = 3 in A + B = 8
A + 3 = 8
Solve for A
A = 5
So, we have:
Hence, the partial fraction decomposition is
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Answer:
- Multiply 5 by 5 to get your first parameter.
- Multiply 6 by 5 to get the denominator, or your second parameter.
- For the second fraction,
, you need to multiply both parameters by 2, similar to before, but we now must use a different number, otherwise, the denominators will not be the same.
- Now, multiply 15 by 2
- The last step is to put these numbers you gathered into fractions. The bigger number always goes on the bottom, referred to as the denominator, while the smaller number, referred to as the numerator, always goes on the top.
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Finally, the problem is solved. Now that the problem is solved, we review what we just learned <em>not through more problems, though.</em>
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<h3>What have we learned?</h3>We learned how to efficiently make fractions' deominators match.
Questions related to this topic? Ask me in the comments box, please!