University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Pascal's triangle is a geometric arrangement of the binomial coefficients in the shape of a triangle. In Pascal's triangle, each number in the triangle is the sum of the two digits directly above it. In Algebra II, we can use the binomial coefficients in Pascal's triangle to raise a polynomial to a certain power.
Using Pascal triangles to expand binomial. So what we're going to do here is introduce Pascal's triangle but before we do that we are going to talk about exactly how these all works. So what we're doing is multiplying out x plus y to a power. Okay, so let's go through a couple of these and see if we can see a pattern. So x plus y to the 0, x plus y is the number 2 is 0, anything to the 0 is just 1. So this just gives us 1, x plus y to the first anything to the first just says as it is, so this just gives us x plus y. Okay x plus y squared, we know this is one of our special formulas so this is just going to give us x squared plus 2xy plus y squared.
Okay, x plus y to the third okay we have to do a little bit more than just remembering this one we'd actually have to multiply it out. Just going to be this times another x plus y it's going to save us the work and we'll just look it with the answer is which is just x cubed plus 3x plus squared y plus 3xy squared plus y cubed. If you don't believe me you can try it out on your own. Okay, so what I want to do now is focus on the coefficients of these terms we have up here. Remembering that whenever those no coefficient there, there's actually a 1, so we have a 1 and 1, 1 and 1 and 1 and 1 on the end as well. Okay so I'm going to write these coefficients out in a triangle form, okay so I'm coming over here what we end up having is 1 from the first, 1, 1 for the second, 1, 2, 1 for the third and 1, 3, 3, 1 for the fourth okay.
This is what is called Pascal's triangle and what this does it just continues, where this is triangle where each number is formed by adding the 2 numbers above it. So if you wanted to continue this on, the 1 is just going to continue down the side, the term that goes underneath the 1 and the 3 is the 4 under the 3 and the 3 add as to get a 6, 3 and 1, 4 and a 1 on the end. So this would actually be the coefficients that we would have if we multiplied x plus y to the fourth. Okay, and another thing we want to look at is we go back over here, the degrees are always adding up to the power that we started with. So for this example here, we have a cubic a third degree, we start with x to the third and our x slowly goes down one degree at a time ending at x is 0, and our y slowly goes up but even here when we x squared y our degrees add up to 3.
Okay, so what that tells me here is if I wanted to fill in the blanks to this one, I know that I'm dealing with a fourth degree so this is going to be x to the fourth, this term is going to be we'll write this outside of the triangle, keep our little triangle clean. This is just going to be x to the fourth and we're going to add 4 degree and x goes down once, this ends being x to the third we need it to still add up too 4 so this is going to be a y going over 1, 6 x degree goes down once we end up with x squared, y degree goes up 1 y squared continue with the pattern x degree goes down 1, y degree goes up 1 lastly finishing with y to the fourth okay. We could continue on to do this for x plus y to the fifth, sixth, seventh you would just need to add on more and more rows.
Obviously this is a little bit easier to use for smaller powers on x plus y once we get to x plus y to the tenth we're going to be writing a lot of rows, as long as you're dealing with smaller powers Pascal's triangles can be really cool resources to help us out.
Unit
Topics in Discrete Math