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

The partial fraction decomposition of the function, $f(x)$, would be:

__First, writing the form of the partial fraction decomposition:__

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

__Second, solving for each of the constants ($A,B,C,D$):__

Clearing the denominator, canceling like terms, and distributing:

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

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

$$4=A(x^2+4)+B(x)(x^2+4)+Cx^3+Dx^2$$

Letting $x=0$:

$$4=A((0)^2+4)+B(0)((0)^2+4)+C(0)^3+D(0)^2$$

$$4=A(4)+0+0+0$$

$$4=A(4)$$

$$A=1$$

Letting $x=2i$:

$$4=((2i)^2+4)+B(2i)((2i)^2+4)+C(2i)^3+D(2i)^2$$

$$4=0+0+8Ci^3+4Di^2$$

$$4+0i=-8Ci-4D$$

Separating real and imaginary terms and solving for constants $C$ and $D$:

Imaginary: $0=-8C$; $C=0$

Real: $4=-4D$; $D=-1$

Solving for $B$ by letting $x=1$ after plugging in all other founds constants and simplifying:

$$4=(x^2+4)+B(x)(x^2+4)-x^2$$

$$4=((1)^2+4)+B(1)((1)^2+4)-(1)^2$$

$$4=(5)+B(5)-(1)$$

$$0=B(5)$$

$$B=0$$

__Next, plugging in the found constants into the form of the partial fraction decomposition:__

$$f(x)=\frac{4}{x^2(x^2+4)}=\frac{A}{x^2}+\frac{B}{x}+\frac{Cx+D}{x^2+4}=\frac{(1)}{x^2}+\frac{(0)}{x}+\frac{(0)x+(-1)}{x^2+4}$$

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

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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__.

__Writing the form of the partial fraction decomposition:__

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

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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__.

__First, factoring and simplifying the denominator:__

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

__Then writing the partial fraction decomposition form for $h(x)$:__

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