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education/math/Partial Fractions.md
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@ -19,8 +19,13 @@ $$ \frac{ax+2a + bx+b}{(x+1)(x+2)} $$
$$ \frac{2x+1}{(x+1)(x+2)} = \frac{ax+2a + bx+b}{(x+1)(x+2)} $$ $$ \frac{2x+1}{(x+1)(x+2)} = \frac{ax+2a + bx+b}{(x+1)(x+2)} $$
5. Notice that the denominator on both sides is equal, meaning you can cancel them out, giving you: 5. Notice that the denominator on both sides is equal, meaning you can cancel them out, giving you:
$$ 2x + 1 = ax + 2a + bx + b $$ $$ 2x + 1 = ax + 2a + bx + b $$
6. Next, group your $x$ values on one side, and your constants on the other side. You'll notice that $ax 6. Next, group your $x$ values on one side, and your constants on the other side. You'll notice that you can factor $ax + bx$, giving you $x(a+b)$.
$$ 2x+1 = $$ $$ 2x+1 = x(a + b) + (2a + b) $$
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:
$$ 2 = a + b $$
$$ 1 = 2a + b $$
## Degree of the numerator is equal ## Degree of the numerator is equal
1. First perform polynomial division. 1. First perform polynomial division.
2. Then find a partial fraction with the remainder 2. Then find a partial fraction with the remainder