QuestionFind the partial fraction decomposition of .
To set it up first write in the form
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Studdy Solution
STEP 1
What is this asking?
We need to break down a complicated fraction into simpler fractions.
Think of it like reversing the addition of fractions!
Watch out!
Don't forget to account for the denominators of the simpler fractions when putting them back together.
STEP 2
1. Factor the Denominator
2. Set up the Decomposition
3. Solve for the Unknowns
4. Verify the Decomposition
STEP 3
Alright, let's **factor** that denominator!
We're looking for two numbers that multiply to and add up to .
Those magic numbers are and .
STEP 4
So, we rewrite the quadratic as .
Now, we can **factor by grouping**.
STEP 5
From the first two terms, we can pull out , giving us .
From the last two terms, we can pull out , giving us .
Notice the common factor !
STEP 6
This means our factored denominator is .
Boom!
STEP 7
Now, we can **set up** our partial fraction decomposition: We're trying to find the mystery numbers and !
STEP 8
To **solve** for and , we'll multiply both sides of the equation by the denominator .
This gives us:
STEP 9
Let's be clever!
If we set , the term with disappears!
Multiplying both sides by and dividing by gives us .
Yes!
STEP 10
Now, let's set to make the term with vanish!
Multiplying both sides by and dividing by gives us .
Awesome!
STEP 11
Let's **verify** our solution!
We found and , so our decomposition is:
STEP 12
We can rewrite this as a single fraction by finding a common denominator: It matches the original expression!
STEP 13
The partial fraction decomposition is:
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