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 $$f(x)=\frac{4}{x^2(x^2+4)}$$.

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

$4=A(x)((x^2)+4)+B((x^2)+4)+Cx+D(x^2)$

Let $x=0$

$4=A(0)((0^2)+4)+B((0^2)+4)+(C(0)+D)(0^2)$

$4=B(4)$
$B=1$

Let $x=4i$

$4=A(4i)((4i^2)+4)+1((4i^2)+4)+(C(4i)+D)(x^2)$

$4=A(4i)((-4)+4)+1((-4)+4)+(C(4i)+D)(-4)$

$4=A(4i)(0)+1(0)+(C(4i)+D)(-4)$

$4=(C(4i)+D)(-4)$

$4=(C(4i)+D)(-4)$

$4=-16iC-4D)$

$4=-4D$

$D=-1$

$-16C=0$

$C=0$

Let $x=1$

$4=A(1)((1^2)+4)+1((1^2)+4)+(0(4i)+(-1))(1^2)$

$4=A5+5-1$

$5=A5+5$

$0=A5$

$A=0$

$\frac{4}{x^2(x^2+4)}$= $\frac{0}{x}+\frac{1}{x^{2}}+\frac{0x+(-1)}{x^{2}+4}$

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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__.

$1=\frac{A}{x+1}+\frac{B}{\left(x+1\right)^{2}}+\frac{C}{\left(x+1\right)^{3}}+\frac{E}{\left(x+1\right)^{4}}+\frac{Fx+G}{x^{2}+1}$

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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__.
$$h(x)=\frac{x^7}{(x^4-16)^2}=\frac{x^7}{(x^2+4)^2(x+2)^2(x-2)^2}$$

$x^{7}=\frac{Ax+B}{x^{2}+4}+\frac{Cx+D}{\left(x^{2}+4\right)^{2}}+\frac{E}{x+2}+\frac{F}{\left(x+2\right)^{2}}+\frac{H}{x-2}+\frac{G}{\left(x-2\right)^{2}}$

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