Find the partial fractions for
(3x^2-7x+12)/(x-2)(x^2-2x+5)
To find the partial fractions of the given rational expression, follow these steps:
Step 1: Factorize the denominator
First, factorize the denominator (x - 2)(x^2 - 2x + 5). The quadratic factor x^2 - 2x + 5 does not factor further unless you want to use complex numbers. So, we'll write it as it is.
The factorization becomes: (x - 2)(x^2 - 2x + 5)
Step 2: Assume the partial fraction decomposition
Write the given rational expression in the form of partial fractions:
(3x^2 - 7x + 12) / (x - 2)(x^2 - 2x + 5) = A / (x - 2) + (Bx + C) / (x^2 - 2x + 5)
Here, A, B, and C are unknown constants that we need to find.
Step 3: Clear the fractions
To remove the fractions in the expression, multiply both sides by the common denominator:
(3x^2 - 7x + 12) = A(x^2 - 2x + 5) + (Bx + C)(x - 2)
Step 4: Solve for unknown constants
Equating the coefficients of like terms on both sides, we can find the values of A, B, and C.
x^2 term: 0A + B = 3x^2 => B = 3
x term: -2A - 2C = -7x
Since no x term on the left-hand side, we can equate the coefficient on the right-hand side to zero:
-2A - 2C = 0 => -2A - 2C = 0 => -2A = 2C
Constant term: 5A - 2C - 2 = 12
5A - 2C = 14
Now, we have two equations:
-2A - 2C = 0 (equation 1)
5A - 2C = 14 (equation 2)
Solve these two equations simultaneously to find the values of A and C.
Step 5: Solve the equations
Multiply equation 1 by -5 to eliminate A:
10A + 10C = 0 (equation 3)
Add equation 2 and equation 3:
10A + 10C + 5A - 2C = 14
15A + 8C = 14
Rearrange the equation:
15A = 14 - 8C
A = (14 - 8C)/15
Substitute the value of A into equation 1:
-2((14 - 8C)/15) - 2C = 0
Simplify the equation:
-28 + 16C - 30C = 0
-14C = 28
C = -2
Substitute the value of C into A:
A = (14 - 8(-2))/15
A = 6/ 15
A = 2/5
So, A = 2/5, B = 3, and C = -2.
Step 6: Write the partial fraction decomposition
Now that we have obtained the values of A, B, and C, we can rewrite the original rational expression using the partial fraction decomposition:
(3x^2 - 7x + 12) / (x - 2)(x^2 - 2x + 5) = 2/5(x - 2) + 3x + (-2) / (x^2 - 2x + 5)
Therefore, the partial fraction decomposition for the given rational expression is:
(3x^2 - 7x + 12) / (x - 2)(x^2 - 2x + 5) = 2/5(x - 2) + 3x - 2 / (x^2 - 2x + 5)