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

1. The fraction f(x) has the partial fraction decomposition form
$$
\frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{x^2+4}
$$

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

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

Let x= 0 to find that A=1
$$4=A(0^2+4)+B(0)(0^2+4)+(C0+D)(0^2)$$
$$4=4A$$
$$1=A$$

Let x= 2i
$$4=A((2i)^2+4)+B(2i)((2i)^2+4)+(C(2i)+D)((2i)^2)$$
$$4= A(-4+4)+ B(2i)(-4+4)-8Ci-4D$$
$$4+0i=-8Ci-4D$$
$$4=-4D$$
$$-1=D$$
$$0i=-8Ci$$
$$0=C$$

Now we can let x=1, plug in our known valus and solve for B.
$$4=A(x^2+4)+B(x)(x^2+4)+(Cx+D)(x^2)$$
$$4=A(1^2+4)+B(1)(1^2+4)+(1C+D)(1^2)$$
$$4=A5+B5+C+D$$
$$4=(1)5+B5+0+-1$$
$$0=5B$$
$$B=0$$

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

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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__.

The fraction g(x) has the partial fraction decomposition form
$$
\frac{A}{(x+1)^4}+\frac{B}{(x+1)^3}+\frac{C}{(x+1)^2}+\frac{D}{(x+1)}+\frac{Ex+F}{(x^2+1)}
$$
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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__.

The fraction h(x) has the partial fraction decomposition form
$$\frac{A}{(x+2)^2}+\frac{B}{(x+2)}+\frac{C}{(x-2)^2}+\frac{D}{(x-2)}+\frac{Ex+F}{(x^2+4)^2}+\frac{Gx+H}{(x^2+4)}$$

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