Partial fraction decomposition
Imagine you have a cake and your friend wants to split it with you. But your friend likes the frosting and you only want the cake part. So you decide to divide the cake into its parts. You take off the frosting part and give it to your friend, and then you are left with the cake part.
Partial fraction decomposition is like dividing the cake into its parts. When we have a complicated fraction, we can break it down into smaller, simpler fractions. We can do this by finding the different parts that make up the fraction.
For example, let's say we have the fraction 2/((x+1)*(x+2)). We can break this into two simpler fractions, one with a denominator of x+1 and the other with a denominator of x+2. We call these simple fractions "partial fractions".
To find the values for these partial fractions, we use algebra. We start by saying that the original fraction is equal to the sum of the two partial fractions:
2/((x+1)*(x+2)) = A/(x+1) + B/(x+2)
We then solve for the variables A and B using some algebra tricks. In this example, we would find that A=1 and B=-1. Now we know that:
2/((x+1)*(x+2)) = 1/(x+1) - 1/(x+2)
So we have broken down the original fraction into two partial fractions. This can be very helpful when trying to integrate or simplify complex expressions.
Partial fraction decomposition is like dividing the cake into its parts. When we have a complicated fraction, we can break it down into smaller, simpler fractions. We can do this by finding the different parts that make up the fraction.
For example, let's say we have the fraction 2/((x+1)*(x+2)). We can break this into two simpler fractions, one with a denominator of x+1 and the other with a denominator of x+2. We call these simple fractions "partial fractions".
To find the values for these partial fractions, we use algebra. We start by saying that the original fraction is equal to the sum of the two partial fractions:
2/((x+1)*(x+2)) = A/(x+1) + B/(x+2)
We then solve for the variables A and B using some algebra tricks. In this example, we would find that A=1 and B=-1. Now we know that:
2/((x+1)*(x+2)) = 1/(x+1) - 1/(x+2)
So we have broken down the original fraction into two partial fractions. This can be very helpful when trying to integrate or simplify complex expressions.