The correct option is B #10; 1x+1x^2+1/x+1 #10; D(x)=x^3+x^2=x^2(x+1) So, 2x^2+2x+1x^2(x+1)=Ax+B/x^2+C/(x+1) So, common denominator in RHS w ...

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Condition for Monotonically Increasing Function

2x2+2x+1/x3+x...

Question

$\frac{2 x^{2} + 2 x + 1}{x^{3} + x^{2}} =$

\frac{1}{x} - \frac{1}{x^{2}} - \frac{1}{x + 1}

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\frac{1}{x} - \frac{1}{x^{2}} + \frac{1}{x + 1}

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\frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x + 1}

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\frac{1}{x} + \frac{1}{x^{2}} - \frac{1}{x + 1}

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Solution

The correct option is B $\frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x + 1}$
$D (x) = x^{3} + x^{2} = x^{2} (x + 1)$
So, $\frac{2 x^{2} + 2 x + 1}{x^{2} (x + 1)} = \frac{A}{x} + B / x^{2} + C / (x + 1)$
So, common denominator in RHS we get
$2 x^{2} + 2 x + 1 = A x (x + 1) + B (x + 1) + C x^{2} - - 1)$
for the value of B substition
$x = 0 \Rightarrow B = 1$
for the value of C keep $x = - 1$
$1 = c \Rightarrow c = 1$
for the value of A keep $x = 1$
$5 = A (2) + 2 B + C$ we know $B = C = 1$
$5 = 2 A + 3 \Rightarrow A = 1$
So, $\frac{1}{x + 1} + \frac{1}{x^{2}} + \frac{1}{(x + 1)}$

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