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Bonus Knowledge
To perform the partial fraction decomposition of
, we first need to factor the denominator. The quadratic
can be factored (if possible) by finding two numbers that multiply to
and add up to
.
After checking potential factors, we find that
.
Now, we can express the fraction as follows:
We set up the partial fraction decomposition:
To find
and
, we multiply through by the denominator
:
Expanding the right side:
Combining like terms gives:
Setting up a system of equations from the coefficients:
Now we can solve for
and
. From the first equation, solve for
:
Substitute
into the second equation:
Now substituting
back into the expression for
:
Thus, we have:
The partial fraction decomposition is:
You can rewrite this as:
So the complete partial fraction decomposition is:
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