Binomial Theorem with Pascal's Triangle
Binomial Theorem - How it Works - Video
Binomial Theorem Expanded
Binomial Theorem Expanded:
Here we have expanded our binomial (a + b). The 1st line has 1 term. The 2nd line has 2 terms. The 3rd line has 3 terms. The 4th line has 4 terms, and so on.
If we take a look at the coefficients in front of each term, the second row has 1 then 1. The 3rd row has 1, 2, 1. The 4th row has 1, 3, 3, 1. Now, instead of distributing a bunch of times, we can use Pascal's triangle to to find the coefficients in any row. With the magic of math, Pascal's Triangle is our friend.
There is a pattern with the variables as well. Can you find?
Binomial Theorem Explanation
Binomial Theorem Explanation:
(a + b)0 = 1 since anything raised to the power of 0 is 1.
The first term in the parenthesis is raised by the power of outside the parenthesis and the second term is raised to the power of 0. Then the power of the first term decreases by 1 while the power of the second term increases by 1
(a + b)1 = 1 * a1 * b0 + 1 * a0 * b1
(a + b)2 = 1 * a2 * b0 + 2 * a1 * b1 + 1 * a0 * b2
(a + b)3 = 1 * a3 * b0 + 3 * a2 * b1 + 3 * a1 * b2 + 1 * a0 * b3
The last piece that we need to talk about are the coefficients. How do we know what they are? Well, we can use Pascal's triangle to find them. Remember, we can find the coefficient by adding the two numbers above.
Properties of the Binomial Theorem
Properties of the Binomial Theorem:
The first property states what we stated already in another way. The power outside the parenthesis tells us how many terms we have after simplifying everything. In the case of (a + b)0. The 0 is n so n + 1 = 0 + 1 = 1 so there is 1 term. If we have (a + b)7. N is 7 so n + 1 = 7 + 1 = 8 so there are 8 terms.
The second property states that the first variable in the parenthesis starts with power outside the parenthesis and decreases each time, while the second variable does the opposite. Finally, when we add the two powers, the result will be same as a power at the beginning.
The third property states how it looks. There must be integers for our powers.
The forth property states another way on how to find the coefficient of the next term.
Example 1
Example 1:
For line 1 - We rewrote the binomial 4 times because of the power 4. If you don't remember the binomial theorem then you can always distribute this way.
For line 2 - We used properties properties 1 and 2 to decrease the variable x from x4 to x0, and increased 3 from 30 to 34.
For line 3 - We simplified the numbers with powers.
For line 4 - We simplified by multiplying the integers.
For line 5 - We wrote the final answer.
Example 2
Example 2:
For line 1 - We rewrote the binomial 4 times because of the power 4. If you don't remember the binomial theorem then you can always distribute this way.
For line 2 - We used properties properties 1 and 2 to decrease the variable x from x4 to x0, and increased 3 from 30 to 34.
For line 3 - We simplified by multiplying the integers.
For line 4 - We simplified by multiplying the integers and wrote the final answer.