Ok u have to put
X^3-8 divided over
X^2(X-1)^3
Into a partial fraction
I need help please
First, check the degree of the numerator and the denominator. If the degree of the numerator is higher than that of the denominator, do a long division to reduce the expression to a proper fraction.
Degree of numerator = 3
Degree of denominator = 2+3 = 5
No long division is required.
Now factor the denominator (already given) and write out the possible fractions of the resulting partial fractions. Use unknown coefficients of A, B, C... for each fraction. For factors with powers (such as x², (x-1)³, etc.), each power is considered a candidate for the resulting partial fraction.
We would end up with
(x³-8)/((x-1)³*x²)
=A/x + B/x² + C/(x-1) + D/(x-1)² + E/(x-1)³ ....(1)
Using a common denominator of x²(x-1)³, the numerator will become:
Ax(x-1)³ + B(x-1)³ + Cx²(x-1)² + Dx²(x-1) + Ex²
Expand the numerator in terms of powers of x and equate to the numerator of the original expression, we should get:
x³-8
=(A+C)x⁴ + (-3A+B-2C+D)x³ + (3A-3B+C-D+E)x² + (-3A+3B)x -B
Comparing coefficients of the constant term:
-B = -8, therefore B=8
Comparing coefficients of the power of x:
-A+3B = 0
A = 3B = 24
Comparing coefficients of x⁴:
A+C = 0
C = -24
Comparing coefficients of x³:
-3A+B-2C+D = 1
D = 1 +3A + B -2C = 17
Finally, solve for E=-7 with the remaining equation (x²).
Substitute the numerical values of A, B, C, D and E into equation (1) to get the decomposition into partial fractions as required.