T2 HBL
Updated: 4/8/2021
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• ﻿﻿﻿﻿﻿﻿﻿﻿n! = n(n-1)(n-2)(n-3) ...
• ﻿﻿﻿﻿﻿Hello class, before we move on to calculus, we have a small topic on Binomial Theorem to cover.﻿﻿﻿﻿
• ﻿﻿BINOMIAL THEOREM﻿﻿
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿(x + y)4 =
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿(x + y)2 =
• ﻿﻿﻿﻿﻿﻿A binomial is a mathematical expression containing two terms.﻿﻿ If raised to an integer power, it can be expanded in this manner. Notice how the power of the first variable decreases by 1 when moving from term to term, while that of the second increases.
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿(x + y)3 =
• ﻿﻿
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿x4 + 4x3y + 6x2y2 + 4xy3 + y4
• ﻿﻿﻿﻿﻿
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿x3 + 3x2y + 3xy2 + y3
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿x2 + 2xy
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿+ y2
• ﻿﻿﻿﻿﻿﻿We can use this array of numbers called the Pascal's triangle, where the row number is the exponent, while the numbers in each row represent the coefficients for the respective terms.
• ﻿﻿﻿﻿11 2 11 3 3 11 4 6 4 11 5 10 10 5 1
• ﻿﻿﻿﻿﻿This enables us to use the nCr notation, to find the coefficient of the rth term in a binomial, starting from 0, among n terms, which is is always 1 more than the exponent.﻿
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿Alternatively, we are able to obtain the coefficients using another method, which involves the concept of factorials, the product of consecutive descending numbers.
• ﻿﻿﻿﻿nCr =
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿n
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿r
• ﻿﻿﻿﻿( )
• ﻿﻿﻿﻿﻿﻿﻿﻿=
• ﻿﻿﻿﻿﻿﻿﻿n!
• ﻿﻿﻿﻿﻿r! (n-r!)
• ﻿﻿﻿With these 2 concepts, we can derive the general term of an expanded binomial. This is particularly useful if we just want to identify a particular term in the whole expression.
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿Tr+1 =
• ﻿﻿﻿﻿In (x + y)n﻿ ,
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿r
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿n
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿( )
• ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿xn-r﻿﻿﻿﻿﻿﻿﻿﻿﻿
• ﻿﻿﻿﻿﻿﻿﻿﻿yr﻿﻿﻿﻿﻿﻿﻿﻿﻿
• ﻿OK class, now take out your A4 paper, we will do some written work, 20 minutes.
• ﻿﻿﻿BINOMIAL THEOREM﻿﻿
• Oh no!!