Use partial fraction to evaluate the following:
(-2x+4)/(x^2+1)(x-1)^2
I will assume your expression is
(-2x+4)/( (x^2+1)(x-1)^2 )
let (-2x+4)/( (x^2+1)(x-1)^2 )
= A/(x-1) + (Bx+C)/(x-1)^2 + (Dx + E)/(x^2 + 1)
then A(x-1)(x^2+1) + (Bx+C)(x^2 + 1) + (Dx+E)(x-1)^2 = -2x + 4
let x = 1
0 + 2B + 2C + 0 = 2 ------> B + C = 1
let x = 0
-A + C + E = 4 ----------> A - C - E = -1
let x = -1
-4A - 2B + 2C - 4D + 4E = 6 ----> 2A + B - C + 2D - 2E = -3
let x = 2
5A + 10B + 5C + 2D + E = 0 ---? 5A + 10B + 5C + 2D + E = 0
let x = -2
-15A - 10B - 5C - 18D + 9E = 8
I ran these 5 equations through a "equation solving" app and got
A = 3/10, B = -4/5, C = 9/5, D = -1, and E = -1/2
take these and form the fractions.
Thank you!
Sure, let's have some fun with partial fractions!
First, let's start with the denominator: (x^2+1)(x-1)^2. We need to factorize it before we can proceed. But don't worry, I won't factorize your hopes and dreams, just the expression!
The quadratic term, x^2+1, is a classic duo that can't be factored further over real numbers, so we'll keep it as it is. However, the binomial term, (x-1)^2, has both comic and real roots. Let's go to the comedy club and factorize this:
(x-1)^2 = (x-1)(x-1)
Now, we have all the ingredients for our partial fraction recipe. Let's assume that our expression can be written as:
(-2x+4)/[(x^2+1)(x-1)^2] = A/(x^2+1) + B/(x-1) + C/(x-1)^2
Now, it's time to find the values of A, B, and C. I hope they don't have stage fright during their performance!
To make things easier, let's find a common denominator for our fractions:
(-2x+4) = A(x-1)^2 + B(x^2+1)(x-1) + C(x^2+1)
Now, let's put on our detective hats and solve the mystery of the missing constants. First, plug in x = 1:
-2(1) + 4 = A(1-1)^2 + B(1^2+1)(1-1) + C(1^2+1)
Simplifying, we get:
2 = A(0) + B(0) + C(2)
2 = 2C
C = 1
Now, let's plug in x = i, where i represents the imaginary unit:
-2(i) + 4 = A(i-1)^2 + B(i^2+1)(i-1) + C(i^2+1)
-2i + 4 = A(i-1)^2 + B(-1+1)(i-1) + C(-1+1)
-2i + 4 = A(i^2-2i+1) + B(0) + C(0)
-2i + 4 = -A(2i) + 1A + 0 + 0
Matching the real and imaginary parts, we get:
-2A = -2i
1A + 0 = 4
From these equations, we find A = i and B = 4.
So, our partial fraction decomposition is:
(-2x+4)/[(x^2+1)(x-1)^2] = (i/(x^2+1)) + (4/(x-1)) + (1/(x-1)^2)
There you have it! The answer is a perfectly balanced combination of real and imaginary humor. Keep solving and keep laughing!
To evaluate the expression (-2x+4)/(x^2+1)(x-1)^2 using partial fractions, we'll follow these steps:
Step 1: Factorize the denominator.
Step 2: Set up the partial fraction decomposition.
Step 3: Solve for the unknown constants.
Step 4: Multiply through by the denominators to clear the fractions.
Step 5: Equate coefficients of like terms.
Step 6: Solve the resulting system of equations.
Step 7: Substitute the values back into the partial fraction decomposition.
Let's begin:
Step 1: Factorize the denominator.
The denominator can be factorized as follows:
(x^2 + 1)(x - 1)^2 = (x^2 + 1)(x - 1)(x - 1)
Step 2: Set up the partial fraction decomposition.
The given expression (-2x + 4)/((x^2 + 1)(x - 1)^2) can be written as:
(A/(x^2 + 1)) + (B/(x - 1)) + (C/(x - 1)^2)
Step 3: Solve for the unknown constants.
To find A, multiply through by (x^2 + 1):
(-2x + 4) = A(x - 1)^2 + B(x^2 + 1)(x - 1) + C(x^2 + 1)
Expanding and simplifying:
-2x + 4 = A(x^2 - 2x + 1) + B(x^3 - x^2 + x - 1) + C(x^2 + 1)
Step 4: Multiply through by the denominators to clear the fractions.
Rearranging the equation above:
-2x + 4 = A(x^2 - 2x + 1) + B(x^3 - x^2 + x - 1) + C(x^2 + 1)
Expanding and collecting like terms:
-2x + 4 = (A + B)x^3 + (C - A + B)x^2 + (-2A + C)x + (A + B + 1)
Step 5: Equate coefficients of like terms.
Comparing coefficients, we have the following equations:
Coefficients of x^3: A + B = 0
Coefficients of x^2: C - A + B = 0
Coefficients of x: -2A + C = -2
Coefficients of the constant term: A + B + 1 = 4
Step 6: Solve the resulting system of equations.
From equation 1, we get B = -A.
Substituting this expression into the other equations:
C - A + (-A) = 0
-2A + C = -2
A - A + 1 = 4
Simplifying:
-C + 2A = 0
-2A + C = -2
1 = 4
Rearranging the equations:
2A - C = 0
-C + 2A = 0
C = 2
2A = 0
A = 0
Step 7: Substitute the values back into the partial fraction decomposition.
Using the values A = 0, B = -A = 0, and C = 2, we can rewrite the given expression as the sum of the partial fractions:
(-2x + 4)/((x^2 + 1)(x - 1)^2) = (0/(x^2 + 1)) + (0/(x - 1)) + (2/(x - 1)^2)
Therefore, the expression (-2x + 4)/((x^2 + 1)(x - 1)^2) can be simplified to:
2/(x - 1)^2
To use partial fraction decomposition, we first need to decompose the given rational function into simpler fractions. The partial fraction decomposition of a rational function involves expressing it as the sum of simpler fractions.
The given rational function is: (-2x+4) / (x^2+1)(x-1)^2.
Step 1: Factorize the denominator.
The denominator (x^2+1)(x-1)^2 can be factorized as follows:
(x^2+1)(x-1)^2 = (x^2+1)(x-1)(x-1).
Step 2: Express the given fraction using partial fractions.
The next step is to express the given fraction in the form of partial fractions. Since the denominator has a quadratic term, we will have to consider the terms in the numerator as linear factors.
Therefore, we express the fraction as:
(-2x+4) / (x^2+1)(x-1)^2 = A/(x^2+1) + B/(x-1) + C/(x-1)^2
Step 3: Find the values of A, B, and C.
To find the values of A, B, and C, we need to clear the fractions by multiplying through by the denominator. After multiplying through, we will equate the coefficients of the corresponding powers of x on both sides.
So, multiplying both sides by (x^2+1)(x-1)^2:
(-2x+4) = A(x-1)^2 + B(x^2+1) + C(x^2+1)(x-1)
Expanding and simplifying:
(-2x+4) = A(x^2-2x+1) + B(x^2+1) + C(x^3 - x^2 + x - 1)
Combining like terms:
-2x + 4 = (A + B)x^2 + (-2A + C)x + (A + B - C)
Now, equating the coefficients of corresponding powers of x:
Coefficient of x^2: A + B = 0
Coefficient of x: -2A + C = -2
Coefficient of x^0 (constant term): A + B - C = 4
Solving these three equations simultaneously will help us find the values of A, B, and C.
Step 4: Solve the system of equations.
From the first equation, we find that A = -B.
Substituting this value into the other equations:
-2(-B) + C = -2 (Substituting A = -B into the second equation)
-B + B - C = 4 (Substituting A = -B into the third equation)
Simplifying the equations:
2B + C = -2 (Equation 1)
-C = 4 (Equation 2)
Solving Equation 2 for C:
C = -4
Substituting C = -4 into Equation 1:
2B - 4 = -2
2B = 2
B = 1
Finally, substituting B = 1 into A = -B:
A = -1
So, we have A = -1, B = 1, and C = -4.
Step 5: Substitute the values of A, B, and C into the partial fraction decomposition equation:
(-2x + 4) / ((x^2 + 1) * (x - 1)^2) = -1 / (x^2 + 1) + 1 / (x - 1) - 4 / (x - 1)^2
Now, the rational function has been expressed as the sum of simpler fractions using partial fraction decomposition.