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education/math/Partial Fractions.md
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@ -4,7 +4,7 @@ This is the "main" method of solving, and the next two headings both focus on ge
1. Factor the bottom. 1. Factor the bottom.
2. Create two fractions, $\frac{a}{p1}$, and $\frac{b}{p2}$, where p1 and p2 are the polynomials you just factored out, and a/b are arbitrary variables 2. Create two fractions, $\frac{a}{p1}$, and $\frac{b}{p2}$, where p1 and p2 are the polynomials you just factored out, and a/b are arbitrary variables
3. Multiply a by p2, and b by p1., giving you: $$\frac{a*p2}{p1} + \frac{b*p1}{p2}$$ 3. Multiply a by p2, and b by p1., giving you: $$\frac{a*p2}{p1} + \frac{b*p1}{p2}$$
4. When you split the 4. Now you can distribute $a$ and $b$, giving you $ax + c$ and $bx + d$. Group and factor. Your equation equals the original equation, so the numerator of the first equals $ax + bx + c + d$.
### Example ### Example
$$ \frac{2x+1}{(x+1)(x+2)} $$ $$ \frac{2x+1}{(x+1)(x+2)} $$
@ -24,7 +24,10 @@ $$ 2x+1 = x(a + b) + (2a + b) $$
7. With the above equation, each side is in the same form. it's $x$ multiplied by a constant ($2$ on the left, and $(a+b)$ on the right, and with a constant of $1$ on the left and $2a + b$) on the right, letting you find the two equations below: 7. With the above equation, each side is in the same form. it's $x$ multiplied by a constant ($2$ on the left, and $(a+b)$ on the right, and with a constant of $1$ on the left and $2a + b$) on the right, letting you find the two equations below:
$$ 2 = a + b $$ $$ 2 = a + b $$
$$ 1 = 2a + b $$ $$ 1 = 2a + b $$
8. The above equations can be solved as a system of equations, giving you 8. The above equations can be solved as a system of equations, giving you:
$$ a=-1,\space b=3 $$
9. Your answer would be:
$$ \frac{-1}{x+1} + \frac{3}{x+2} $$
## Degree of the numerator is equal ## Degree of the numerator is equal
1. First perform polynomial division. 1. First perform polynomial division.