Vandermonde polynomial

Ok kiddo, imagine you have a bunch of numbers, let's say 1, 2, and 3. We can use these numbers to create something called a Vandermonde polynomial.

Now, a polynomial is just a fancy word for a math formula that has powers of variables. For example, something like x^2 + 2x + 1 is a polynomial.

So, how do we make a Vandermonde polynomial out of our numbers? Well, we take the first number (1) and raise it to different powers. Let's say we raise it to the power of 0, 1, and 2. This gives us:

1^0, 1^1, 1^2

Simplifying that, we get:

1, 1, 1

Why did we do that? Well, we're going to use those numbers to make a polynomial.

Next, we do the same thing with the second number (2). We raise it to the same powers as before (0, 1, and 2) to get:

2^0, 2^1, 2^2

Simplifying, we get:

1, 2, 4

Lastly, we do the same thing with the third number (3). We raise it to the same powers as before (0, 1, and 2) to get:

3^0, 3^1, 3^2

Simplifying, we get:

1, 3, 9

We now have our three sets of numbers:

1, 1, 1
1, 2, 4
1, 3, 9

And we can use them to make a Vandermonde polynomial! The formula for it looks like this:

f(x) = a0 + a1x + a2x^2

But instead of those letters (a0, a1, and a2), we will use our sets of numbers.

This gives us:

f(x) = 1 + 2x + x^2

And that, my dear little one, is how you make a Vandermonde polynomial!