Partial Fraction with Cubic Denominator :
Here we are going to see some examples on decomposition of partial fractions with cubic denominator.
Example 1 :
Resolve the following rational expressions into partial fractions.
(6x2 - x + 1)/(x3 + x2 + x + 1)
Solution :
Let us use synthetic division to factorize the cubic denominator.
6x2 - x + 1 = A(x2 + 1) + (Bx + C)(x + 1)
6x2 - x + 1 = Ax2 + A + Bx2 + Bx + Cx + C
Equating the coefficient of x2, x and constant terms
6 = A + B ----(1)
-1 = B + C ----(2)
1 = A + C ----(3)
(1) - (2)
A + B - (B + C) = 6 + 1
A - C = 7 ----(4)
(3) + (4) ==> A + C + A - C = 1 + 7
2A = 8
A = 4
Apply A = 4 in (1) 4 + B = 6 B = 6 - 4 B = 2 |
Apply B = 2 in (2) 2 + C = -1 C = - 1 - 2 C = -3 |
Hence the solution is
Example 2 :
Resolve the following rational expressions into partial fractions.
(2x2 + 5x - 11)/(x2 + 2x - 3)
Solution :
(2x2 + 5x - 11)/(x2 + 2x - 3)
= 2 + [(x - 5)/(x + 3)(x - 1)]
x - 5 = A/(x + 3) + B(x - 1)
x - 5 = A(x - 1) + B(x + 3)
If x = 1 -4 = B(4) 4B = -4 B = -1 |
If x = -3 -8 = A(-4) A = 8/4 A = 2 |
Hence the solution is
Example 3 :
Resolve the following rational expressions into partial fractions.
(7 + x)/(1 + x)(1 + x2)
Solution :
(7 + x)/(x + 1)(x2 + 1) = A/(x + 1) + (Bx+C)/(x2 + 1)
7 + x = A(x2 + 1) + (Bx + C)(x + 1)
0 = A + B ---(1)
1 = B + C ---(2)
7 = A + C ---(3)
(1) - (2)===>
A - C = -1 ---(4)
(3) + (4) ==>
2A = 6 ===> A = 3
B = -3
C = A + 1 ==> 3 + 1 ==> 4
C = 4
Hence the solution is
3/(x + 1) + (4 - 3x)/(x2 + 1)
After having gone through the stuff given above, we hope that the students would have understood, how to decompose rational expressions with cubic denominator into partial fractions.
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