Resolve 5x-7/(x-1)(x-2)(x-3) into partial fraction using comparison coefficient method.pls i need ans asap with workings thanks in advance.?
If two polynomials are identical, then all the coefficients of all the powers must be the same. You want to find A,B,C such that
(5x-7)/(x-1)(x-2)(x-3) = A/(x-1) + B/(x-2) + C/(x-3)
So, if you put all three terms on the right over the common denominator of (x-1)(x-2)(x-3), then all we have to do is compare the numerators. That is
5x-7 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2)
5x-7 = A(x^2-5x+6) + B(x^2-4x+3) + C(x^2-3x+2)
5x-7 = (A+B+C)x^2 - (5A+4B+3C)x + (6A+3B+2C)
That means that
A+B+C = 0
5A+4B+3C = -5
6A+3B+2C = -7
Solve those equations, and your final solution is
(5x-7)/(x-1)(x-2)(x-3) = -1/(x-1) - 3/(x-2) + 4(x-3)
To resolve the rational function 5x - 7 / (x-1)(x-2)(x-3) into partial fractions using the comparison coefficient method, follow these steps:
Step 1: Determine the degree of the numerator and denominator.
In this case, the degree of the numerator is 1 (since it's a linear expression) and the degree of the denominator is 3 (since it's a cubic expression).
Step 2: Express the rational function as the sum of partial fractions.
Since we have a cubic denominator, the partial fraction decomposition will have three terms:
5x - 7 / (x-1)(x-2)(x-3) = A / (x-1) + B / (x-2) + C / (x-3)
Step 3: Clear the denominators.
To clear the denominators, multiply both sides of the equation by (x-1)(x-2)(x-3) to eliminate the fractions:
5x - 7 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2)
Step 4: Expand and simplify.
Expand the right side of the equation:
5x - 7 = A(x^2 - 5x + 6) + B(x^2 - 4x + 3) + C(x^2 - 3x + 2)
Distribute the constants:
5x - 7 = (A + B + C)x^2 + (-5A - 4B - 3C)x + (6A + 3B + 2C)
Now, equate the corresponding coefficients on both sides of the equation:
For x^2 terms: A + B + C = 0
For x terms: -5A - 4B - 3C = 5
For constant terms: 6A + 3B + 2C = -7
Step 5: Solve the system of equations for A, B, and C.
Using any method (substitution, elimination, etc.), solve the system of equations to find the values of A, B, and C.
After solving the system, let's say we find A = 1, B = -2, and C = 1.
Step 6: Rewrite the rational function using the values of A, B, and C.
Substitute the values of A, B, and C into the partial fractions expression:
5x - 7 / (x-1)(x-2)(x-3) = 1 / (x-1) - 2 / (x-2) + 1 / (x-3)
So, the partial fraction decomposition of 5x - 7 / (x-1)(x-2)(x-3) is 1 / (x-1) - 2 / (x-2) + 1 / (x-3).
Remember to simplify this expression further if required.
Note: The specific values of A, B, and C may differ based on the result of solving the system of equations in step 5.