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Question: How do you express \[\dfrac{1}{{{(s+1)}^{2}}}\] in partial fractions?...

How do you express 1(s+1)2\dfrac{1}{{{(s+1)}^{2}}} in partial fractions?

Explanation

Solution

The given expression first needed to be expressed in a correct partial form type. After that simplifying it and then finding the value of the unknowns we assumed as numerator and then substituting it to get the partial fraction. At the end you will find that the given expression is already expressed in partial form.

Complete step-by-step answer:
This question belongs to the concept of partial fraction decomposition more specifically irreducible quadratic denominators.
In order to solve the question, let us get a brief introduction of the concept of partial fractions and its decomposition.
Partial fractions are simply breaking the polynomial into simpler form in order to get the solution.
We have to perform few steps in order to express a given question into partial fractions. First, we have to factorize the denominator after that we write the fractions with one of the factors for each of the denominators. Also, we don’t know the value of numerator thus we assign it unknown values. after that you equate them. Then we multiply the common denominator and get rid of the denominator. Then you multiply, solve and find the values of unknown by creating the system of equations and solving them. After that putting the values of unknown in the equation you formed. Thus, you expressed the equation in the partial fraction form.
Now let us solve the question
In the question we have
1(s+1)2\dfrac{1}{{{(s+1)}^{2}}}
This type of partial fraction will be of the type
1(s+1)2=As+1+B(s+1)2−−(1)\dfrac{1}{{{(s+1)}^{2}}}=\dfrac{A}{s+1}+\dfrac{B}{{{(s+1)}^{2}}}--(1)
Here we have expressed the numerator in the Left-hand side in terms of unknown
Now on simplifying we get
1(s+1)2=As+A+B(s+a)2\dfrac{1}{{{(s+1)}^{2}}}=\dfrac{As+A+B}{{{(s+a)}^{2}}}
Now we will compare the coefficients of the terms on both the side
Here, we can see that the coefficient is on left hand side is zero and for the constant term it is one
Thus, we get A=0,A+B=1A=0,A+B=1.
Using the above two values we get A=0,B=1A=0,B=1
Now substitute the value of A and B in equation (1)
1(s+1)2=0s+1+1(s+1)2\dfrac{1}{{{(s+1)}^{2}}}=\dfrac{0}{s+1}+\dfrac{1}{{{(s+1)}^{2}}}
Or we can say that
1(s+1)2=1(s+1)2\dfrac{1}{{{(s+1)}^{2}}}=\dfrac{1}{{{(s+1)}^{2}}}
This means that the given expression is already expressed in its partial form.
Hence, we can conclude that 1(s+1)2\dfrac{1}{{{(s+1)}^{2}}} is already in its partial form.

Note: Since this question is a bit tricky as the given expression is already in its partial form. So, make sure you don’t get confused at the end. Solve the equation for finding the value of the unknowns. Make sure you use the right substitution. Keep an idea of the method used to solve the question.