I need these questions..

Here's the answer  to (1)

http://web2.0calc.com/questions/what-is-the-coefficient-of-nbsp-nbsp-in-nbsp_2

2. Use the Binomial Theorem to write $(3+sqrt{5})^4$ in the form $a + bsqrt{5}$ for some positive integers a and b.

(3 + sqrt(5) ) ^4   =

3^4   +  (4)(3)^3 * sqrt(5)  + (6)(3)^2* [sqrt(5)]^2  +  (4)(3)(sqrt(5))^3  +  (sqrt(5))^4   = 

81  + 108sqrt (5)    +  54 *5   +  60sqrt(5)   +  25  =

81  +  54 *5  + 25 + 108sqrt(5)  + 60sqrt(5)  =

81  +  270  + 25   + 168sqrt(5)  =

376  + 168sqrt(5)

3.Find the constant term in the expansion of \[\Big(z - \frac{2}{\sqrt{z}}\Big)^9.\]

[ z  - 2/sqrt(z)]^ 9

The constant term  occurs   at    C(9,6) (z)^3 [2/ sqrt(z)]^6   =  84 z^3 [ 64/ z^3]  =

84 * 64   =

5376

4a.Expand $\left(x^2+\frac{1}{x}\right)^3$. (Write the terms with higher degree first, so for example an x^2 term would come before x or 1/x.)

[ x^2  +  1/x] ^3   =

(x^2)^3  +  (3)(x^2)^2 (1/x)  + (3)(x^2)(1/x)^2   + (1/x)^3   =

x^6   +  3x^3   + 3    + (1/x)^3  =

x^6  + 3x^3  + (1/x)^3  + 3

4b. Find the constant term in the expansion of \($\Big(x^2+\frac{1}{x}\Big)^4$\)

There is no constant term  in this expansion.......

5a. For what positive integers n does $\(\left(x^2+\frac{1}{x}\right)^n\)$ have a nonzero constant term ??

This will occur  when n  = 3p   for p ≥ 1

5b. For the values of n that you found in part (a), what is that constant term?

The term that will procuce a constant  is given by :   C(3p,2p)(x^2)^p*(1/x)^(2p) 

Could you explain how you got the formula for the constant term?

Could you explain questions 5a and 5b please?

Yes, CPhill could you please explaing how you go it?

Why did you use a p for 5a explanation?

Can it be any variable?

aops much

The constant term in 5b is 0.

The constant term in 5b is 0.