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### Course: Precalculus>Unit 9

Lesson 3: The binomial theorem

# Expanding binomials

Sal expands (3y^2+6x^3)^5 using the binomial theorem and Pascal's triangle. Created by Sal Khan.

## Want to join the conversation?

• This problem is a bit strange to me. Sal says that "We've seen this type problem multiple times before." I haven't. I must have missed several videos along the way. Can someone point me in the right direction? Thank's very much.
• What does Sal mean by 5 choose 1?
• 5C1 or 5 choose 1 refers to how many combinations are possible from 5 items, taken 1 at a time. That means that 5 choose 1 is 5.

This topic is related to probability (permutations and combinations). You might want to check out that topic if you don't understand.

(My Source: http://www.statisticshowto.com/5-choose-3-5c3-figuring-combinations/)
• how do we solve this type of problem when there is only variables and no numbers?
• The only way I can think of is (a+b)^n where you would generalise all of the possible powers to do it in, but thats about it, in all other cases you need to use numbers
• Wow. This makes absolutely zero sense whatsoever.
• Its just a specific example of the previous binomial theorem where a and b get a little more complicated. Rather than figure out ALL the terms, he decided to hone in on just one of the terms. Sometimes in complicated equations, you only care about 1 or two terms.
• how do you know if you have to find the coefficients of x6y6
• It would be included in the question; eg. consider the binomial expansion of (a+b)^n. Find the coefficient of b^(n-2).
• at , what are the exclamation marks (factorials)? do they symbolize a number?
• If n is a positive integer, then n! means "n factorial", which is defined as the product of the positive integers from 1 to n inclusive (for example, 4! = 1*2*3*4 = 24). Furthermore, 0! is defined as 1.
The exclamation mark is itself not a number, but it can be thought of as a function (or unary operation).

Have a blessed, wonderful day!
• how do you do it when the equation is (a-b)^7? When the sign is negative, is there a different way of doing it? If there is a new way, why is that?
• The only difference is the 6x^3 in the brackets would be replaced with the (-b), and so the -1 has the power applied to it too. Odd powered brackets would therefore give negative terms and even powered brackets would gve a positive term.
• can someone please tell or direct me to the proof/derivation of the binomial theorem.
it would be much appreciated
• The coefficient of x^2 in the expansion of (1+x/5)^n is 3/5, (i) Find the value of n
(ii) With this value of n, find the term independent of x in the expansion (1+x/5)^n(2-3/x)^2
• sounds like we want to use pascal's triangle and keep track of the x^2 term. We can skip n=0 and 1, so next is the third row of pascal's triangle.

1 2 1 for n = 2
the x^2 term is the rightmost one here so we'll get 1 times the first term to the 0 power times the second term squared or 1*1^0*(x/5)^2 = x^2/25 so not here.

1 3 3 1 for n = 3
Squared term is second from the right, so we get 3*1^1*(x/5)^2 = 3x^2/25 so not here

1 4 6 4 1 for n = 4
Squared term is the third from the right so we get 6*1^2*(x/5)^2 = 6x^2/25

1 5 10 10 5 1 for n = 5
Squared term is fourth from the right so 10*1^3*(x/5)^2 = 10x^2/25 = 2x^2/5 getting closer.

1 6 15 20 15 6 1 for n=6
Fifth from the right here so 15*1^4*(x/5)^2 = 15x^2/25 = 3x^2/5 There we are.

So n has to equal 6 Now you can use this line of the triangle to find the term where the x part is brought to the 0 power.

If you're unfamiliar with pascal's triangle this probably doesn't make sense, and I'd be happy to explain it.