QuestionWrite the form of the partial fraction decomposition of the rational expression given below. Do not solve for the constants.
Studdy Solution
STEP 1
What is this asking?
We need to rewrite this big fraction as the sum of smaller, simpler fractions, but we don't need to figure out what the numbers on top of those smaller fractions are!
Watch out!
Don't forget to factor the bottom of the fraction first!
Also, remember we're not solving for the mystery numbers on top of the smaller fractions, just setting things up!
STEP 2
1. Factor the Denominator
2. Decompose the Fraction
STEP 3
Alright, let's **factor** the bottom of our fraction, which is .
We're looking for two numbers that add up to **9** and multiply to **18**.
Think, think, think... **6** and **3**!
Perfect!
STEP 4
So, we can rewrite as .
This is super important because it tells us what the smaller fractions will look like!
STEP 5
Now, here's the fun part!
We're going to rewrite our big fraction as the sum of two smaller fractions.
Since the bottom of our original fraction factors into and , our smaller fractions will have these as their denominators.
STEP 6
We don't know what the numbers on top of these smaller fractions are yet, so we'll use letters, like *A* and *B*, to represent them.
This gives us:
STEP 7
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