In my math book it says: $${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({{\mathtt{s}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}\left({\mathtt{s}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)\right)}}$$=$${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({\mathtt{s}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({{\mathtt{s}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{s}}}}$$

But I just can't wrap my head around it! How do they get from left to right?

Can somebody please help me? :)

Thanks so much if you can :)

Guest Jun 30, 2015

#2**+10 **

This is known as a partial fraction decomposition....let's look at this......

-1 / [ s^2 * ( s + 1) ] ....we can split this up as

-1 / [ s^2 * (s + 1)] = A / s + B /s^2 + C / (s + 1) ...where A, B and C are coefficients to be determined

Multiply both sides by [ s^2 * (s + 1)] ....so we have......

-1 = As(s + 1) + B (s + 1) + Cs^2 simplify (factor) the right side like so .....

-1 = (A + C)s^2 + (A + B)s + B then, equating coefficients sets up the following system

B = -1

A + B = 0 which implies that A = 1

A + C = 0 which implies that C = -1 so we have

-1 / [ s^2 * (s + 1)] = 1 / s - 1 /s^2 - 1 / (s + 1) which is essentially the same as your result

If you're unfamiliar with this technique, here's a good primer........http://www.purplemath.com/modules/partfrac.htm

CPhill
Jun 30, 2015

#1**0 **

"How do they get from left to right?"

You are trying wrong way.. it's simple to get from right to left.

- 1 / (s + 1) - 1 / s^2 + 1/s = - s^2 / (s^2(s+1)) + (- s - 1)/(s^2(s+1)) + (s^2 + s)/(s^2(s+1))

= - 1 / (s^2(s+1))

Guest Jun 30, 2015

#2**+10 **

Best Answer

This is known as a partial fraction decomposition....let's look at this......

-1 / [ s^2 * ( s + 1) ] ....we can split this up as

-1 / [ s^2 * (s + 1)] = A / s + B /s^2 + C / (s + 1) ...where A, B and C are coefficients to be determined

Multiply both sides by [ s^2 * (s + 1)] ....so we have......

-1 = As(s + 1) + B (s + 1) + Cs^2 simplify (factor) the right side like so .....

-1 = (A + C)s^2 + (A + B)s + B then, equating coefficients sets up the following system

B = -1

A + B = 0 which implies that A = 1

A + C = 0 which implies that C = -1 so we have

-1 / [ s^2 * (s + 1)] = 1 / s - 1 /s^2 - 1 / (s + 1) which is essentially the same as your result

If you're unfamiliar with this technique, here's a good primer........http://www.purplemath.com/modules/partfrac.htm

CPhill
Jun 30, 2015