Express (3x^3+2x^2+2x-3)/((x^2+2)(x+1)^2) in partial fraction. Help any1 plz ASAP. Thanks a million
To express the rational expression (3x^3+2x^2+2x-3)/((x^2+2)(x+1)^2) in partial fraction, follow these steps:
Step 1: Factor the denominator completely.
(x^2+2) can't be factorized any further since it is an irreducible quadratic.
(x+1)^2 can be factorized as (x+1)(x+1).
Step 2: Write the expression as the sum of partial fractions with unknown constants in the numerator.
A/(x^2+2) + (Bx + C)/(x+1) + (Dx + E)/(x+1)^2 = (3x^3+2x^2+2x-3)/((x^2+2)(x+1)^2)
Step 3: Clear fractions by multiplying both sides by ((x^2+2)(x+1)^2).
A(x+1)^2 + (Bx + C)(x^2+2) + D(x^2+2) + E(x+1) = 3x^3+2x^2+2x-3
Step 4: Expand everything and collect like terms.
A(x^2+2x+1) + B(x^3+2x) + C(x^2+2) + D(x^2+2) + E(x+1) = 3x^3+2x^2+2x-3
Step 5: Combine like terms on both sides.
(A + C + D)x^2 + (2A - E)x + (A + 2C + D + E) = 3x^3+2x^2+2x-3
Step 6: Equate the coefficients of each power of x on both sides.
For x^3, 0 = 3, so A = 0.
For x^2, 1 = 2, so A + C + D = 0.
For x, 0 = 2, so 2A - E = 0.
For the constant term, -3 = A + 2C + D + E.
Step 7: Solve this system of equations to find the values of the constants A, B, C, D, and E.
From the equations obtained in step 6:
A = 0,
C = -1,
D = 1,
E = 0,
and B remains unknown.
Therefore, the expression can be expressed as:
0/(x^2+2) + (-x)/(x+1) + (x+1)/(x+1)^2
Simplifying further, we get:
-x/(x+1) + 1/(x+1)
So, the expression (3x^3+2x^2+2x-3)/((x^2+2)(x+1)^2) can be written as:
-x/(x+1) + 1/(x+1)
I hope this explanation helps you understand how to express a rational expression in partial fraction decomposition.