Binomial theorem

Question

Find the coefficient of the indicated term in each expansion

(4a +2b)^8 , 5th term

(a- 2 √3^6 , 3rd term

Answer 1

You need to understand the pattern...

See watch, the first term, the first expression has expoenent 8

2nd term the first expression has expoenent 7

3rd term the first expression has expoenent 6

etc etc..

X^8 + (8 choose 1) X^7 Y + (8 Choose 2) X^6 Y^2 ... etc.. (8 choose 7) X Y^7 + Y^8

the 2nd expression, the exponent goes UP:

2nd expression is NOT in the first term

2nd expression has expoenent 1 in the 2nd term

2nd expression has expoenent 2 in the 3rd term

etc etc

following this pattern, the 5th term is (8 choose 4) X^4 Y^4 where

X = 4a and Y = 2b

[8*7*6*5*4!/ (2! * 4!)] ( 4a)^4 ( 2b)^4 =

8*7*6*5/2 ( 256a^4)(16 b^4) =

(4*7*6*5) (256) (16) a^4 b^4

3,440,640 a^4 b^4

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N=6

In the 3rd term, the first expression has exponent 4...

so the general form of the term is (8 choose 2) x^4 y^2

(8 choose 2) = 8*7*6!/ 2! = 8*7*6/2 = 8*7*3 = 8*21 = 168

168 a^4 ( 2 rad3)^2 =

168 a^4 4*3 =

2016 a^4