Welcome to MooMooMath.
In this video I'm talking about binomial expansion What does binomial expansion mean?
It means if I have a binomial like x+5 to the 4th power
I have different ways to multiply that.
The first thing I'm going to do is it to the fourth power?
What does it mean to raise the binomial to the fourth power?
It means x+4 times x+4 times x+4 times x+4 I have to multiply all of these together
What I can do is foil, and multily the first two terms then the last two terms, and then
multiply this answer.
There is another way to do this, and that is to use Pascal's triangle to expand this.
What is pascal's triangle?
Pascal's triangle are the coefficients when you raise binomials to different powers.
If I have a binomial a+b to zero power is 1.
If I have a+b to the first it is 1 and 1 Now if I raise it to the 2nd power
I have which is a+b2 = a+b xa+b It works out that my coefficients are 1 2 1
What you can see is that the numbers are generated by adding the layer before.
To draw the next layer you get 1 3 3 1
If I raise it to the fourth you get 1 4 6 4 and 1
You can see each layer I'm adding I'm adding the coefficients in the triangle.
What I have done is created this fourth layer so those will be the coefficients.
In this example I take x+4 to the fourth The coefficients will be 1 4 6 4 1
What I will do is take the first term and raise it to the fourth term and the first
term will descend so the second term is raised to the fourth power,the third term to the
6th fourt to the 4th.
You can see the first term has desending powers.
So I get 4 to 1st 4 to the second 4 to the third and 4 to the fourth, and I do this for
my 5 terms.
Now each column will represent the term in my binomial answer.
First term will be x to the 4th The second term will be 4 times 4 times x
cubed The next term will be 16 times 6 equals 96
squared.
The next term is 4 cubed times 4 which is 256 and 1 times x to the zero is one and 4
to the 4th is 256 So this is how I would expand x+4 using Pascal's