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Question

The partial fractions of $fraction numerator 1 over denominator x to the power of 3 end exponent left parenthesis x plus 2 right parenthesis end fraction$ are

$\frac{1}{8 x} - \frac{1}{4 x^{2}} + \frac{1}{2 x^{3}} - \frac{1}{8 (x + 2)}$
$\frac{1}{8 x} + \frac{1}{4 x^{2}} + \frac{1}{2 x^{3}} - \frac{1}{8 (x + 2)}$
$\frac{1}{8 x} - \frac{1}{4 x^{2}} - \frac{1}{2 x^{3}} + \frac{1}{8 (x + 2)}$
$\frac{1}{8 x} + \frac{1}{4 x^{2}} + \frac{1}{2 x^{3}} + \frac{1}{8 (x + 2)}$

hint Hint:

The partial fraction decomposition is writing a rational expression as the sum of two or more partial fractions. The following steps are helpful to understand the process to decompose a fraction into partial fractions.

Step-1: Factorize the numerator and denominator and simplify the rational expression, before doing partial fraction decomposition.

Step-2: Split the rational expression as per the formula for partial fractions. P/((ax + b)² = [A/(ax + b)] + [B/(ax + b)²]. There are different partial fractions formulas based on the numerator and denominator expression.

Step-3: Take the LCM of the factors of the denominators of the partial fractions, and multiply both sides of the equation with this LCM.

Step-4: Simplify and obtain the values of A and B by comparing coefficients of like terms on both sides.

Step-5: Substitute the values of the constants A and B on the right side of the equation to obtain the partial fraction.

The correct answer is: $fraction numerator 1 over denominator 8 x end fraction minus fraction numerator 1 over denominator 4 x to the power of 2 end exponent end fraction plus fraction numerator 1 over denominator 2 x to the power of 3 end exponent end fraction minus fraction numerator 1 over denominator 8 left parenthesis x plus 2 right parenthesis end fraction$

Given : $fraction numerator 1 over denominator x to the power of 3 end exponent left parenthesis x plus 2 right parenthesis end fraction$
Step-1: Split the rational expression as per the formula for partial fractions.
$rightwards double arrow fraction numerator 1 over denominator x cubed left parenthesis x plus 2 right parenthesis end fraction space equals space A over x space plus space B over x squared space plus space C over x cubed space plus space fraction numerator D over denominator left parenthesis x space plus space 2 right parenthesis end fraction$
Step-2: Take the LCM of the factors of the denominators of the partial fractions, and multiply both sides of the equation with this LCM.
. $rightwards double arrow fraction numerator 1 over denominator x cubed left parenthesis x plus 2 right parenthesis end fraction space equals space fraction numerator A left parenthesis x squared right parenthesis left parenthesis x space plus 2 right parenthesis space plus space B left parenthesis x right parenthesis left parenthesis x space plus 2 right parenthesis space plus space C left parenthesis x plus 2 right parenthesis space plus space D left parenthesis x cubed right parenthesis over denominator x cubed left parenthesis x plus 2 right parenthesis end fraction C a n c e l l i n g space t h e space d e n o m i n a t o r s space o n space b o t h space s i d e s rightwards double arrow 1 space equals space A left parenthesis x squared right parenthesis left parenthesis x space plus 2 right parenthesis space plus space B left parenthesis x right parenthesis left parenthesis x space plus 2 right parenthesis space plus space C left parenthesis x plus 2 right parenthesis space plus space D left parenthesis x cubed right parenthesis$

Step-3: Simplify and obtain the values of A , B , C and D by comparing coefficients of like terms on both sides
$rightwards double arrow 1 space equals space A left parenthesis x cubed space plus 2 x squared right parenthesis space plus space B left parenthesis x squared space plus 2 x right parenthesis space plus space C left parenthesis x plus 2 right parenthesis space plus space D left parenthesis x cubed right parenthesis rightwards double arrow 1 space equals x cubed left parenthesis A space plus D right parenthesis space plus x squared left parenthesis B space plus 2 A right parenthesis space plus space x left parenthesis 2 B space plus C right parenthesis space plus space 2 C C o m p a r i n g space a n d space e q u a t i n g rightwards double arrow A space plus space D space equals space 0 rightwards double arrow B space plus space 2 A space equals space 0 rightwards double arrow 2 B space plus space C space equals space 0 rightwards double arrow 2 C space equals space 1 space rightwards double arrow C space equals 1 half S u b s t i t u t i n g space t h e space v a l u e space o f space C rightwards double arrow B space equals space minus 1 fourth space comma space A space equals space 1 over 8 space a n d space D space equals space minus 1 over 8$

Step-4: Substitute the values of the constants A, B , C and D on the right side of the equation to obtain the partial fraction.

$rightwards double arrow fraction numerator 1 over denominator x cubed left parenthesis x plus 2 right parenthesis end fraction space equals space A over x space plus space B over x squared space plus space C over x cubed space plus space fraction numerator D over denominator left parenthesis x space plus space 2 right parenthesis end fraction$
$rightwards double arrow fraction numerator 1 over denominator x cubed left parenthesis x plus 2 right parenthesis end fraction space equals space fraction numerator 1 over denominator 8 x end fraction space minus space fraction numerator 1 over denominator 4 x squared end fraction space plus space 2 1 over x cubed space minus space fraction numerator 1 over denominator 8 left parenthesis x space plus space 2 right parenthesis end fraction$

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parallel

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