Solve the partial fraction (X-1)(X+1)/(X+2)(X^2-3X+1).
To solve the partial fraction decomposition, we first factor the denominator:
X^2 - 3X + 1 = (X - φ)(X - ψ)
where φ and ψ represent the roots of the quadratic equation.
The roots of the quadratic equation can be found using the quadratic formula:
X = (-(-3) ± √((-3)^2 - 4(1)(1))) / (2(1))
= (3 ± √(5)) / 2
So, the roots of the quadratic equation are (3 + √5) / 2 and (3 - √5) / 2.
Therefore, the factored form of the denominator is:
(X + 2) (X - (3 + √5) / 2) (X - (3 - √5) / 2)
Now, let's write the partial fraction decomposition:
(X - 1)(X + 1) / ((X + 2)(X^2 - 3X + 1))
= A / (X + 2) + (B(X - (3 + √5) / 2) + C(X - (3 - √5) / 2)) / (X^2 - 3X + 1)
To find the values of A, B, and C, we can multiply both sides of the equation by the denominator:
(X - 1)(X + 1) = A(X^2 - 3X + 1) + (B(X - (3 + √5) / 2) + C(X - (3 - √5) / 2))(X + 2)
Expanding the equation and combining like terms, we get:
X^2 - 1 = A X^2 + (-3A + B + C) X + (A + (2B - (3 + √5) / 2) + (2C - (3 - √5) / 2))
Now, we need to match the coefficients of the corresponding powers of X on both sides of the equation.
For X^2:
A = 1
For X^1:
-3A + B + C = 0
For X^0:
A + (2B - (3 + √5) / 2) + (2C - (3 - √5) / 2) = -1
Simplifying the equations, we have:
A = 1
-3 + B + C = 0
2B + 2C - 3 + √5 - 3 - √5 = -1
Simplifying further:
B + C = 3
2B + 2C - 6 = -1
2B + 2C = 5
Simplifying again:
B + C = 3
B + C = 2.5
We have two conflicting equations, which means there is no solution for this partial fraction decomposition.
(x-1)(x+1)/((x+2)(x^2-3x+1))
= 3/(11(x+2)) + (8x-7)/(11(x^2-3x+1))
I apologize for the mistake in my previous response. You are correct, the partial fraction decomposition of (x-1)(x+1)/((x+2)(x^2-3x+1)) can be written as:
(x-1)(x+1)/((x+2)(x^2-3x+1)) = 3/(11(x+2)) + (8x-7)/(11(x^2-3x+1))
This can be obtained by finding the values of A, B, and C through a similar process as previously described.
To solve the partial fraction decomposition of the given expression, (X-1)(X+1)/(X+2)(X^2-3X+1), you can follow these steps:
Step 1: Factorize the denominator:
The denominator, (X+2)(X^2-3X+1), can be factored by splitting the middle term of the quadratic equation, X^2-3X+1, as follows:
X^2-3X+1 = (X-2)(X-1)
So the factored form of the denominator is:
(X+2)(X-2)(X-1)
Step 2: Write the partial fraction decomposition form:
The given expression can be written in the form of partial fraction decomposition as:
(X-1)(X+1)/(X+2)(X^2-3X+1) = A/(X+2) + B/(X-2) + C/(X-1)
Step 3: Solve for the unknown coefficients A, B, and C:
To find the values of A, B, and C, you can equate the numerators of both sides of the equation. This will give you a system of equations to solve for A, B, and C.
(X-1)(X+1) = A(X-2)(X-1) + B(X+2)(X-1) + C(X+2)(X-2)
Expand both sides of the equation:
X^2 - 1 = A(X^2 - 3X + 2) + B(X^2 + X - 2) + C(X^2 - 4)
Now, collect like terms on the right-hand side of the equation:
X^2 - 1 = (A + B + C)X^2 + (-3A + B)X + (2A - 2B - 4C)
Equate the coefficients of corresponding powers of X:
For the term X^2: 1 = A + B + C
For the term X: 0 = -3A + B
For the constant term: -1 = 2A - 2B - 4C
Now you have a system of three linear equations in A, B, and C. You can solve this system of equations to find the values of A, B, and C.