Question

s+2/(s+4)(s+1) partial fraction​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Resolve in to partial fraction :-

$\sf \: \dfrac{s + 2}{(s + 4)(s + 1)}$

$\large\underline{\bold{ANSWER-}}$

$\rm :\longmapsto\:Let \: \dfrac{s + 2}{(s + 4)(s + 1)} = \dfrac{a}{s + 4} + \dfrac{b}{s + 1} - - (1)$

• Multiply by (s + 4)(s + 1) both sides, we get

$\rm :\longmapsto\:s + 2 = a(s + 1) \: + \: b(s + 4)$

• Substituting 's = - 1' in the given equation, we get

$\rm :\longmapsto\:( - 1 + 2) = 0 + b( - 1 + 4)$

$\rm :\implies\:b \: = \: \dfrac{1}{3} - - (2)$

Now,

• Substituting 's = - 4' in the given equation, we get

$\rm :\longmapsto\: - 4 + 2 = a( - 4 + 1) + 0$

$\rm :\longmapsto\: - 2 = - 3a$

$\rm :\implies\:a = \dfrac{2}{3} - - (3)$

Now,

• Substitute values of a and b in equation (1), we get

$\rm :\implies\: \boxed{ \tt\dfrac{s + 2}{(s + 4)(s + 1)} = \dfrac{2}{3(s + 4)} + \dfrac{1}{3(s + 1)}}$

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What is Partial Fraction?

• This method is used to decompose a given rational expression into simpler fractions.

Rules to break the term in Partial Fraction :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf factor \: in \: denominator & \bf term \: in \: partial \: fraction \\ \frac{\qquad \qquad\qquad\qquad}{} & \frac{\qquad \qquad\qquad\qquad}{} \\ \sf ax + b & \sf\displaystyle \frac{A}{{ax + b}} \\ \\ \sf {\left( {ax + b} \right)^2} & \sf \displaystyle \frac{{{A_1}}}{{ax + b}} + \frac{{{A_2}}}{{{{\left( {ax + b} \right)}^2}}} \\ \\ \sf {ax}^{2} + bx + c & \sf \displaystyle \frac{{Ax + B}}{{a{x^2} + bx + c}} \end{array}} \\ \end{gathered}$