Math 182 Miniproject 1 Partial Fractions.md
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Math 182 Miniproject 1 Partial Fractions
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**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 $$
\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+1}+\frac{Fx+G}{(x^2+1)^2}.
$$
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__Problem 1.__
Find the partial fraction decomposition of the function $$f(x)=\frac{4}{x^2(x^2+4)}$$.

$f\left(x\right)=\frac{4}{x^{2}\left(x^{2}+4\right)}=\frac{A}{x^{2}}+\frac{B}{x}+\frac{\left(Cx+D\right)}{x^{2}+4}$

$4=A\left(x^{2}+4\right)+Bx\left(x^{2}+4\right)+\left(Cx+D\right)\left(x^{2}\right)$

Let $x=0$

$4=A\left(0^{2}+4\right)+B\left(0\right)\left(0^{2}+4\right)+\left(C\left(0\right)+D\right)\left(0^{2}\right)$

$A\left(4\right)=4$
$A=1$

Let $x=2i$

$4=1\left(\left(2i\right)^{2}+4\right)+B\left(2i\right)\left(\left(2i\right)^{2}+4\right)+\left(C\left(2i\right)+D\right)\left(2i^{2}\right)$

$4=0+0+\left(C\left(2i\right)+D\right)\left(-4\right)$

$4=-8Ci-4D$
$4=-4D$ & $0=-8C$

$D=-1$ & $C=0$

Let $x=1$

$4=1\left(1^{2}+4\right)+B\left(1\right)\left(1^{2}+4\right)+\left(\left(0\right)\left(1\right)-1\right)\left(1\right)^{2}$

$4=5+B\left(5\right)-1$

$B=0$

$\int_{ }^{ }\frac{1}{x^{2}}dx+\int_{ }^{ }\frac{0}{x}dx+\int_{ }^{ }-\frac{1}{x^{2}+4}dx$

$\int_{ }^{ }\frac{1}{x^{2}}dx$ $=-\frac{1}{x}+c$

$\int_{ }^{ }\frac{0}{x}dx$ $=c$

$\int_{ }^{ }-\frac{1}{x^{2}+4}dx$ $=-\frac{1}{2}\arctan\left(\frac{x}{2}\right)+c$

$f\left(x\right)=\frac{4}{x^{2}\left(x^{2}+4\right)}=-\frac{1}{x}-\frac{1}{2}\arctan\left(\frac{x}{2}\right)+c$

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__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__.

$$g(x)=\frac{1}{(x+1)^4(x^2+1)}$$ $$=\frac{A}{\left(x+1\right)^{4}}+\frac{B}{\left(x+1\right)^{3}}+\frac{C}{\left(x+1\right)^{2}}+\frac{D}{\left(x+1\right)}+\frac{\left(Ex+F\right)}{\left(x^{2}+1\right)}$$

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__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__.

$$h(x)=\frac{x^7}{(x^4-16)^2}$$ $$=\frac{x^{7}}{\left[\left(x^{2}-4\right)\left(x^{2}+4\right)\right]^{2}}$$

$$=\frac{x^{7}}{\left(x+2\right)^{2}\left(x-2\right)^{2}\left(x^{2}+4\right)^{2}}$$

$$=\frac{A}{\left(x+2\right)^{2}}+\frac{B}{\left(x+2\right)}+\frac{C}{\left(x-2\right)^{2}}+\frac{D}{\left(x-2\right)}+\frac{\left(Ex+F\right)}{\left(x^{2}+4\right)^{2}}+\frac{\left(Gx+11\right)}{\left(x^{2}+4\right)}$$
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