Can you please help me with the following questions please, I don't understand them. I know the general rule about them.
Write the partial fraction decomposition of the rational expression.
(x^2+4x-1)/(x^2+3)^2
(4x^3+4x^2)/(x^2+5)^2
Usually partial fractions have two or more different factors in the denominator.
Yours is not the typical style
Here is a nice page that explains the regular kind and the method in very simple language.
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/partial.html
Sure, I can help you with the partial fraction decomposition of the given rational expressions.
To find the partial fraction decomposition, we need to decompose the given rational expression into simpler fractions with denominators that are linear, irreducible quadratic, or repeated linear factors.
Let's start with the first expression: (x^2+4x-1)/(x^2+3)^2.
Step 1: Determine the factorization of the denominator (x^2+3)^2.
The denominator is already factored correctly, so we can move to the next step.
Step 2: Set up the partial fraction decomposition.
The given expression (x^2+4x-1)/(x^2+3)^2 can be written as:
(x^2+4x-1)/(x^2+3)^2 = A/(x^2+3) + B/(x^2+3)^2
Step 3: Find the values of A and B.
To determine the values of A and B, we need to clear the fractions by multiplying through by the common denominator, which is (x^2 + 3)^2. This gives us:
x^2 + 4x - 1 = A(x^2 + 3) + B
Expanding the right side, we get:
x^2 + 4x - 1 = Ax^2 + 3A + B
Matching coefficients of like terms on both sides, we have:
1 = A (for the x^2 terms)
4 = 3A (for the x terms)
Solving this system of equations, we find A = 1 and B = 1.
Step 4: Write the partial fraction decomposition.
Now that we have found the values of A and B, we can write the partial fraction decomposition:
(x^2+4x-1)/(x^2+3)^2 = 1/(x^2+3) + 1/(x^2+3)^2
Similarly, let's now solve the second expression: (4x^3+4x^2)/(x^2+5)^2.
Step 1: Determine the factorization of the denominator (x^2+5)^2.
The denominator is already factored correctly, so we can move to the next step.
Step 2: Set up the partial fraction decomposition.
The given expression (4x^3+4x^2)/(x^2+5)^2 can be written as:
(4x^3+4x^2)/(x^2+5)^2 = A/(x^2+5) + B/(x^2+5)^2
Step 3: Find the values of A and B.
To determine the values of A and B, we need to clear the fractions by multiplying through by the common denominator, which is (x^2 + 5)^2. This gives us:
4x^3 + 4x^2 = A(x^2 + 5) + B
Expanding the right side, we get:
4x^3 + 4x^2 = Ax^2 + 5A + B
Matching coefficients of like terms on both sides, we have:
4 = A (for the x^2 terms)
0 = 5A + B (for the x^3 terms)
From the second equation, we can solve for B in terms of A and substitute the value of A from the first equation. This gives us B = -5(4) = -20.
Therefore, A = 4 and B = -20.
Step 4: Write the partial fraction decomposition.
Now that we have found the values of A and B, we can write the partial fraction decomposition:
(4x^3+4x^2)/(x^2+5)^2 = 4/(x^2+5) - 20/(x^2+5)^2
I hope this explanation helps you understand how to find the partial fraction decomposition of rational expressions. If you have any further questions, feel free to ask!