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title: Math 182 Miniproject 1 Partial Fractions

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Math 182 Miniproject 1 Partial Fractions.md
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Math 182 Miniproject 1 Partial Fractions
===

**Overview:** In this project we explore more advanced partial fraction decomposition than we covered during class.

**Prerequisites:** Section 5.5 of _Active Calculus_ and a strong background in solving systems of linear equations.

For this miniproject we will need to know the general theory of partial fraction decompositions. We can rewrite a rational function $\frac{f(x)}{g(x)}$ by factoring $g(x)$ and looking at the powers of unique factors.



| Factor of $g(x)$ | Term in partial fraction | 
| -------- | -------- | 
| $ax+b$    | $\frac{A}{ax+b}$    | 
| $(ax+b)^k$    | $\frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+\cdots+\frac{A_k}{(ax+b)^k}$    | 
| $ax^2+bx+c$    | $\frac{Ax+B}{ax^2+bx+c}$    | 
| $(ax^2+bx+c)^k$    | $\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\cdots+\frac{A_kx+B_k}{(ax^2+bx+c)^k}$    | 

If the degree of $f(x)$ is greater than or equal to the degree of $g(x)$, then we have to do long division before finding the partial fraction decomposition. 

__Example.__ The fraction $$\frac{4x^4+34x63+71x^2-32x-128}{x^2(x+4)^3}$$ has a partial fraction decomposition of the form $$
\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+4}+\frac{D}{(x+4)^2}+\frac{E}{(x+4)^3}.
$$

__Example.__ The fraction $$\frac{x^6+x^4+x^3-x^2-1}{x^3(x^2+1)^2}$$ has a partial fraction decomposition of the form $$
\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+1}+\frac{Fx+G}{(x^2+1)^2}.
$$
___

__Problem 1.__
Find the partial fraction decomposition of the function $$f(x)=\frac{4}{x^2(x^2+4)}$$.


Solution goes here.
$$f(x)=\frac{4}{x^2(x^2+4)}=\frac{A}{(x)}+\frac{B}{x^2}+\frac{Cx+D}{(x^2+4)}$$


$${4}={Ax}{(x^2+4)}+{B}{(x^2+4)}+{(Cx+D)}{(x^2)}$$
$${x=0,4=4B, B=1}$$
$${x=2i,-4=-8iC+4D}$$
$$-1+0i=-2iC+D, D=-1$$
$${x=i,4=-iA-iC+4iA+4}$$
$${0i=3iA-iC}$$
$${0i=3iA; Oi=Ci}$$
$${A = 0, B = 1, C = 0, D = -1}$$
$$f(x)=\frac{0}{(x)}+\frac{1}{x^2}+\frac{0-1}{(x^2+4)}$$
$$f(x)=\frac{1}{x^2}+\frac{-1}{(x^2+4)}$$
__Problem 2.__
For the function $$g(x)=\frac{1}{(x+1)^4(x^2+1)}$$ write the form of the partial fraction decomposition. __Do not find the full partial fraction decomposition__.


Solution goes here.
$$g(x)=\frac{A}{(x+1)}+\frac{B}{(x+1)^2}+\frac{C}{(x+1)^3}+\frac{D}{(x+1)^4}+\frac{Ex+F}{(x^2+1)}
$$
___

__Problem 3.__
For the function $$h(x)=\frac{x^7}{(x^4-16)^2}$$ write the form of the partial fraction decomposition. __Do not find the full partial fraction decomposition__.


Solution goes here.


_$$h(x)=\frac{x^7}{(x^2+4)^2(x^2-4)^2}$$_
$$h(x)=\frac{x^7}{(x^2+4)^2(x+2)^2(x-2)^2}$$
$$h(x)=\frac{A}{(x-2)}+\frac{B}{(x-2)^2}+\frac{C}{(x+2)}+\frac{D}{(x+2)^2}+\frac{Ex+F}{(x^2+4)}+\frac{Gx+H}{(x^2+4)^2}$$

To submit this assignment click on the __Publish__ button. Then copy the url of the final document and submit it in Canvas.



