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 $$f(x)=\frac{4}{x^2(x^2+4)}$$.
__Find the partial fraction decomposition form__

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

__Multiply both sides of the equation by the LCD__

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

__Distribute and group like terms together__

$$Ax^3+4Ax+Bx^2+4B+Cx^3+Dx^2=4$$ $$Ax^3+Cx^3+Bx^2+Dx^2+4Ax+4B=4$$ $$(A+C)x^3+(B+D)x^2+(4A)x+4B=4$$

__Let $x=0$__

$$A(0)^3+D(0)^3+B(0)^2+D(0)^2+4A(0)+4B=4$$
$$4B=4 \implies B=1$$

__Let $x=2i$__
$$(A+C)(2i)^3+(1+D)(2i)^2+(4A)(2i)=0$$ $$(A+C)(-8i)+(1+D)(-4)+(4A)(2i)=0$$ $$-8Ai-8Ci-4-4D+8Ai=0$$ $$-8Ci-4D-4=0$$
__Solve for C & D through a system of linear equations__

$$-8Ci-4D=0i+4$$ $$-8Ci=0i \implies C=0$$ $$-4D=4 \implies D=-1$$

__Solve for A after plugging in B, C, & D__

$$Ax(x^2+4)+B(x^2+4)+(Cx+D)x^2=4$$ $$Ax(x^2+4)+x^2+4-x^2=4$$ $$Ax(x^2+4)=0 \implies A=0$$

A=0
B=1
C=0
D=-1

__Rewrite the partial fraction decomposition using values found for A, B, C, & D__

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

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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__.

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

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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__.

__Factor the denominator because it is not in the form of $(ax+b)^k$ or $(ax^2+bx+c)^k$__
$$\frac{x^7}{((x^2+4)(x^2-4))^2}=\frac{x^7}{((x^2+4)(x+2)(x-2))^2}=\frac{x^7}{(x^2+4)^2(x+2)^2(x-2)^2}$$

__Find the partial fraction decomposition__
$$\frac{A}{(x+2)}+\frac{B}{(x+2)^2}+\frac{C}{(x-2)}+\frac{D}{(x-2)^2}+\frac{Ex+F}{x^2+4}+\frac{Gx+H}{(x^2+4)^2}$$

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