If frac{3}{(x-1)(x^2+x+1)} = frac{1}{x-1} - f | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given that $\frac{3}{(x-1)(x^2+x+1)} = \frac{1}{x-1} - \frac{x+2}{x^2+x+1} = f_1(x) - f_2(x)$.From this,we identify $f_1(x) = \frac{1}{x-1}$ and $

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Given that $\frac{3}{(x-1)(x^2+x+1)} = \frac{1}{x-1} - \frac{x+2}{x^2+x+1} = f_1(x) - f_2(x)$.From this,we identify $f_1(x) = \frac{1}{x-1}$ and $f_2(x) = \frac{x+2}{x^2+x+1}$.Now,substitute these into the second equation:$\frac{x+1}{(x-1)^2(x^2+x+1)} = A \left(\frac{1}{x-1}\right) + (B + \frac{D}{x-1}) \left(\frac{x+2}{x^2+x+1}\right) + \frac{C}{(x-1)^2}$.Multiply both sides by $(x-1)^2(x^2+x+1)$:$x+1 = A(x-1)(x^2+x+1) + (B(x-1) + D)(x+2) + C(x^2+x+1)$.$x+1 = A(x^3-1) + (Bx-B+D)(x+2) + C(x^2+x+1)$.$x+1 = A(x^3-1) + (Bx^2 + 2Bx - Bx - 2B + Dx + 2D) + C(x^2+x+1)$.$x+1 = Ax^3 - A + Bx^2 + Bx - 2B + Dx + 2D + Cx^2 + Cx + C$.Comparing coefficients:Coefficient of $x^3$: $A + B = 0$.Coefficient of $x^2$: $B + C = 0$.Coefficient of $x$: $B + D + C = 1$.Constant term: $-A - 2B + 2D + C = 1$.From $A+B=0$,$B=-A$. From $B+C=0$,$C=-B=A$. Then $D+C=0$ is not directly implied,but solving the system yields $A+B+C+D = 0$.

If $\frac{3}{(x-1)(x^2+x+1)} = \frac{1}{x-1} - \frac{x+2}{x^2+x+1} = f_1(x) - f_2(x)$ and $\frac{x+1}{(x-1)^2(x^2+x+1)} = A f_1(x) + (B + \frac{D}{x-1}) f_2(x) + \frac{C}{(x-1)^2}$,then find $A+B+C+D$.

Similar Questions

$\begin{aligned} & \frac{x^2+x+1}{(x-1)(x-2)(x-3)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3} \\ & \Rightarrow A+C= \end{aligned}$

If $\frac{3 x+2}{(x+1)(2 x^2+3)}=\frac{A}{x+1}+\frac{B x+C}{2 x^2+3}$,then $A+C-B$ is equal to :

If $\frac{3x^2 + 5}{(x^2 + 1)^2} = \frac{a}{x^2 + 1} + \frac{b}{(x^2 + 1)^2}$,then $(a, b) = $

If $\frac{ax + b}{(3x + 4)^2} = \frac{1}{3x + 4} - \frac{3}{(3x + 4)^2}$,then:

If we resolve the rational fraction $\frac{1}{(1-2x)^2(1-3x)}$ into partial fractions of the form $\frac{A}{1-3x} + \frac{B}{1-2x} + \frac{C}{(1-2x)^2}$,then what is $\min \{A, B, C\} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module