Question

Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients.$\frac{t^4+t^2+1}{(t^2+4)(t^2+2)^2}$

Answer 1

The form of the partial fraction decomposition is $\frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}$

How to decompose the function?

The fraction is given as:

$\frac{t^4 + t^2 + 1}{(t^2 + 4)(t^2 + 2)^2}$

To decompose the above, we make use of the following rule:

$\frac{px^n + qx^m +....+ r}{(x^2 + a)(x^2 + b)^2} = \frac{Ax + B}{(x^2 + a)} + \frac{Cx + D}{(x^2 + b)} + \frac{Ex + F}{(x^2 + b)^2}$

So, we have:

$\frac{t^4 + t^2 + 1}{(t^2 + 4)(t^2 + 2)^2} = \frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}$

Hence, the form of the partial fraction decomposition is $\frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}$

Read more about partial fraction decomposition at:

brainly.com/question/2516522

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