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How do you write the partial fraction decomposition of the rational expression #(x^2)/(x+1)^3#?

Answer 1

Christian Antunez

#1/(x+1)-2/(x+1)^2+1/(x+1)^3#

#x^2/(x+1)^3=A/(x+1)+B/(x+1)^2+C/(x+1)^3#

#x^2=A(x+1)^2+B(x+1)+C#

#x^2=Ax^2+2Ax+A+Bx+B+C#

#x^2=x^2(A)+x(2A+B)+1(A+B+C)#

Thus, #{(A=1),(2A+B=0),(A+B+C=0):}#

Solve to see that #{(A=1),(B=-2),(C=1):}#

Therefore,

#x^2/(x+1)^3=1/(x+1)-2/(x+1)^2+1/(x+1)^3#

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Answer 2

Sadie Alexander

To write the partial fraction decomposition of the rational expression ( \frac{x^2}{(x+1)^3} ), first, express the denominator as a product of linear factors:

[ (x+1)^3 = (x+1)(x+1)(x+1) ]

Then, the partial fraction decomposition takes the form:

[ \frac{x^2}{(x+1)^3} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} ]

To determine the values of ( A ), ( B ), and ( C ), you can either equate coefficients or use another appropriate method. After finding the values of ( A ), ( B ), and ( C ), substitute them back into the original expression to obtain the partial fraction decomposition.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.