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## Precalculus

### Course: Precalculus>Unit 9

Lesson 3: The binomial theorem

# Intro to the Binomial Theorem

The Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. The larger the power is, the harder it is to expand expressions like this directly. But with the Binomial theorem, the process is relatively fast! Created by Sal Khan.

## Want to join the conversation?

• Why is 0! = 1 ?
Isn't factorial just a fancy way of saying multiply all the previous numbers together, like 4! = 1*2*3*4 ?
• Caleb Joshua's response makes sense. I give him a credit.
My response will be based on the study of patterns that result from factorials of consecutive numbers starting from 1, leaving that of 0! as a task to read from the pattern. Pay a closer attention to the computations inside brackets. I hope you will follow. This is how it goes.

1!=1 ?
2!=2 (2*1!)
3!=6 (3*2!)
4!=24 (4*3!)
5!=120 (5*4!)
6!=720 (6*5!)
7!=5040 (7*6!)
Note: Start reading the brackets from bottom going up to see the pattern.
Would you please check the result for 1!. If you read the pattern of computations in brackets, you would note that 1!=1= 1*0!. Then whats 0!? 1 is a multiplicative identity of integers (from Abstract Algebra). Multiplying a number by 1 equals the same number. So if 1!=1 and 1=1*0!, then 0! equals the one on the left of the equation 1=1*0!. Thus 0!=1.
• At , is that n "choose" k? I've seen this notation before and have wondered what it meant. Is there a video that shows where this comes from?
• Thank you! This video was very helpful... but I do have another question that was not addressed in it. What would I do if I have to expand a binomial with two coefficients?
For example, I've been trying to solve this: (3x + 2y)^5
I've expanded it to this: 3x^5 + 5 * 3x^4 2y + 10 3x^3 2y^2 + 10 3x^2 2y^3 + 5 * 3x 2y^4 + 2y^5
... but, I don't know where to go at this point.
How would I simplify this binomial even further??
• I think he probably addresses that in the more detailed videos, as this was just an introduction to this concept.
(1 vote)
• At , where did the K come from in (a+b) to the n power?
• Is there any easier, quicker way to do the binomial expression, besides using this long equation? Because the equation is a lot to remember!
• DamianmusK
Try it out! Expand a binomial to the powers 1,2,3,4,etc.
Then verify the numbers and you will be intrigued and may remember it.
Psychological studies show that elaborate memory is better than rote memory( relating STM data to past experiences helps). In this case, you will realise that learning this equation is better than solving binomials as your brain will associate solving with the pain of expanding the terms

Hope this helps!
@Rock11
• At , what does Sal mean by N choose K? Is that a matrix?
• Not really. A matrix would be indicated by multiple columns and/or rows of numbers, all enclosed by brackets ( these -----> [ ] ) that appear to be "stretched" vertically to enclose the entire ends. n choose k is indicated by a number or variable on top of another number or variable, enclosed by parentheses (as opposed to brackets). n is the top, k is the bottom. "n choose k" is a combination, the number of possible distinct ways to choose k objects (order being irrelevant) from a set of n objects. Does that make sense?
Sorry. I think I might have been a little too long-winded.
• What does that E mean? Is it Sigma? I do 't know. It's the first time I've seen it.
• Is there a video where we can learn more about factorials, and how to figure them out?
• The symbol after the equals sign (with n above and k = 0 below) - how does this symbol work? I.e. does the symbol represent an algorithm that sums all of the values gained from iterating between k and n? Hope that makes sense. Basically I can see the way it works but I want to understand the mechanics of it.