Binomial Expansion: Solve Example & Get Help

[RMHAZARD]

hi i have a whole page of bionomial expansion homework from as teacher and have never encountered it before here is an example of one of my questions could someone please show me how its done.

example:
find the coefficient of x (cubed) in the expansion of (2 + 3x) (to the power of 5)

i am completely stuck,

thanks in advance.

 

[ghetto_bird25]

x (cubed) in the expansion of (2 + 3x) (to the power of 5)

well that's kinda easy actually, just think of
(2 + 3x)^{5} as (2 +3x)(2 +3x)(2 +3x)(2 +3x)(2 +3x)
so just multiply out using foil...looks like its going to take a while but I am assuming its just for practice and to get u into the mood of foil for other questions about functions

 

[kbaumen]

To solve (a+b)^{n}, Pascal's triangle may help you:
1 n=0
1 1 n=1
1 2 1 n=2
1 3 3 1 n=3
1 4 6 4 1 n=4
1 5 10 10 5 1 n=5
1 6 15 20 15 6 1 n=6
1 7 21 35 35 21 7 1 n=7
1 8 28 56 70 56 28 8 1 n=8

I've shown the first 9 lines, however, you can continue writing lines indefinitely. Every number is the sum of upper two. For example in n=3 line, 1 = 0 + 1, 2 = 1 + 1, 1 = 1 + 0. In n=4 line 1 = 0 + 1, 3 = 1 + 2, 3 = 2 + 1, 1 = 1 + 0 and so on.

Numbers in these lines coefficients for expansion of (a+b)^{n}.

(a+b)^{n} can be expanded to Ca^{n}+Ca^{n-1}b+Ca^{n-2}b^{2}+...+Ca^{2}b^{n-2}+Cab^{n-1}+Cb^{n} where C are the coefficients from line n.

For example:
(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}

To expand (2+3x)^{5} we must take the coefficients from n=5 line =>
(2+3x)^{5}=2^{5}+5*2^{4}*3x+10*2^{3}*(3x)^{2}+10*2^{2}*(3x)^{3}+5*2*(3x)^{4}+(3x)^{5} = 32+270x+720x^{2}+1080x^{3}+810x^{4}+243x^{5}

If I calculated correctly, than from this you can see that the coefficient of x cubed is 1080. I think that expanding the binomial this way is much easier than multiplying.

I hope that helps.

P.S. If you see the triangle with a straight angle, than look http://en.wikipedia.org/wiki/Pascal_triangle" .

 

[robert Ihnot]

You can get at this quicker through binominal coefficients: (3X+2)^5, the coefficient on x^3 will be 5!/(3!2!) = 10, but you have to consider the three on x and the 2 as well.