Partial fraction
When we have a big fraction with a bunch of numbers on top and bottom, it can be really hard to understand. But sometimes, we can break that big fraction into smaller fractions, and that makes it easier to figure out what it means.
So, let's say we have a fraction like this: 10 / (x+3)(x+5). That might look confusing, but we can turn it into two smaller fractions like this:
10 / ((x+3)(x+5)) = A / (x+3) + B / (x+5)
A and B are just two new numbers we don't know yet. But we can find out what they are by doing some math. We multiply both sides of the equation by the common denominator, which is (x+3)(x+5). That gives us:
10 = A(x+5) + B(x+3)
Now we can solve for A and B by substituting in some values for x. For example, if we put x = -3, the second term on the right side of the equation goes away, and we end up with:
10 = A(-3+5)
10 = 2A
A = 5
If we put x = -5 instead, we get:
10 = B(-5+3)
10 = -2B
B = -5
So now we know that:
10 / ((x+3)(x+5)) = 5 / (x+3) - 5 / (x+5)
That might look a little confusing too, but it's actually a lot easier to work with. We can use these two smaller fractions to solve problems and understand things better. And that's what partial fractions are all about!
So, let's say we have a fraction like this: 10 / (x+3)(x+5). That might look confusing, but we can turn it into two smaller fractions like this:
10 / ((x+3)(x+5)) = A / (x+3) + B / (x+5)
A and B are just two new numbers we don't know yet. But we can find out what they are by doing some math. We multiply both sides of the equation by the common denominator, which is (x+3)(x+5). That gives us:
10 = A(x+5) + B(x+3)
Now we can solve for A and B by substituting in some values for x. For example, if we put x = -3, the second term on the right side of the equation goes away, and we end up with:
10 = A(-3+5)
10 = 2A
A = 5
If we put x = -5 instead, we get:
10 = B(-5+3)
10 = -2B
B = -5
So now we know that:
10 / ((x+3)(x+5)) = 5 / (x+3) - 5 / (x+5)
That might look a little confusing too, but it's actually a lot easier to work with. We can use these two smaller fractions to solve problems and understand things better. And that's what partial fractions are all about!