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
Sure, I can help you with the partial fraction decomposition of both rational expressions. To decompose a rational expression into partial fractions, follow these steps:
Step 1: Factor the denominator
Factor the denominator of the rational expression as much as possible.
Now, let's work on the first rational expression:
1. (x^2+4x-1)/(x^2+3)^2
Step 2: Identify the degree of the numerator and denominator.
In this case, the numerator and denominator both have a degree of 2.
Step 3: Determine the general form of the partial fractions.
For a denominator raised to a power of 2, the general form of the partial fractions will be:
A/(x^2+3) + B/(x^2+3)^2
Step 4: Find the values of A and B.
To find the values of A and B, we need to combine the partial fractions back together and equate the numerator coefficients.
(x^2+4x-1)/(x^2+3)^2 = A/(x^2+3) + B/(x^2+3)^2
Multiplying through by (x^2+3)^2:
(x^2+4x-1) = A(x^2+3) + B
Expanding everything:
x^2 + 4x - 1 = Ax^2 + 3A + B
Step 5: Equate the coefficients.
Equate the coefficients of the corresponding powers of x:
x^2 coefficient:
1 = A
x coefficient:
4 = 0 (since the x term is absent on the right side)
Constant term:
-1 = 3A + B
From the equation 1 = A, we find A = 1.
Substituting A = 1 into -1 = 3A + B, we can solve for B:
-1 = 3(1) + B
-1 = 3 + B
B = -4
Therefore, the partial fraction decomposition of (x^2+4x-1)/(x^2+3)^2 is:
(x^2+4x-1)/(x^2+3)^2 = 1/(x^2+3) - 4/(x^2+3)^2
Now let's move on to the second rational expression:
2. (4x^3+4x^2)/(x^2+5)^2
Step 2: Identify the degree of the numerator and denominator.
In this case, the numerator has a degree of 3, and the denominator has a degree of 2.
Step 3: Determine the general form of the partial fractions.
For a denominator raised to a power of 2, the general form of the partial fractions will be:
A/(x^2+5) + B/(x^2+5)^2
Step 4: Find the values of A and B.
Following the same procedure as before, we equate the numerator coefficients:
(4x^3+4x^2)/(x^2+5)^2 = A/(x^2+5) + B/(x^2+5)^2
Multiplying through by (x^2+5)^2:
(4x^3+4x^2) = A(x^2+5) + B
Expanding everything:
4x^3 + 4x^2 = Ax^2 + 5A + B
Step 5: Equate the coefficients.
Equate the coefficients of the corresponding powers of x:
x^3 coefficient:
4 = 0 (since the x^3 term is absent on the right side)
x^2 coefficient:
4 = A
x coefficient:
0 = 5A
Constant term:
0 = B
From the equation 4 = A, we find A = 4. Also, from 0 = B, we find B = 0.
Therefore, the partial fraction decomposition of (4x^3+4x^2)/(x^2+5)^2 is:
(4x^3+4x^2)/(x^2+5)^2 = 4/(x^2+5)