## anonymous 5 years ago what is (2x+3) times itself 10 times?

1. anonymous

x= 5

2. anonymous

you mean $(2x+3)^{10}$?

3. anonymous

perhaps this is the answer or perhaps you are being asked to expand?

4. anonymous

alright well the original question is " find the coefficient of x^6 in the expansion of (2x+3)^10

5. anonymous

ahh that is a different story

6. anonymous

because writing out the entire expansion is a pain.

7. anonymous

Remember this formula for co-efficient during expansion: 1 1 = Power 1 1 2 1 = Power 2 (Retain the 1st number, add 1st and 2nd number put as 2nd, retain 2nd number as 3rd) 1 3 3 1 = Power 3 (Retain the 1st number, add 1st and 2nd number put as 2nd, add 2nd and 3rd as 3rd, retain 3rd number as 4th) 1 4 6 4 1 = Power 4 1 5 10 10 5 1 = Power 5 Hence the coefficient of x³ is: 10*(-2)³ *3²= -80 * 9 = -720 Note: Interestingly: (1) The terms for power 1 is 2, for power2 it is 3, for power n the terms are n + 2 (2) The sum of the coefficients for power 1 is: 1+1 = 2 = 2¹, for power 2, 1+2+1 = 4 = 2², power 3 is 1+3+3+1=2³,..., for power n, it is 2^n

8. anonymous

well thats what i have to do is find the coefficient of x^6 in the expansion of (2x+3)^10

9. anonymous

ohh kk

10. anonymous

$\dbinom{10}{6}(2x)^6$

11. anonymous

^^ idk what that means....

12. anonymous

The coefficient of x^5 is -8064 The x^6 has a coefficient of -13608

13. anonymous

$\dbinom{10}{6}=\frac{10\times 9 \times 8\times 7}{4\times 3\times 2}=210$

14. anonymous

$2^6=64$

15. anonymous

wait wat that like wrong

16. anonymous

maybe . i am tired. why? i thought it was 64*210=13440

17. anonymous

oh carp. i am wrong. i forgot $3^4$ that is there too!

18. anonymous

im sorry i have no idea what the H ur talking about

19. anonymous

term with $x^6$ looks like $\dbinom{10}{6}(2x^6)(3^4)$

20. anonymous

if you expand $(a+b)^n$ the terms look like $\dbinom{n}{k}a^kb^{n-k}$ here n= 10 k=6 a = 2x b=3

21. anonymous

ohhh okay haha thanks

22. anonymous

so you get $\dbinom{10}{6}(2x)^63^4$

23. anonymous

$\dbinom{10}{6}=210$ $2^6=64$ $3^4=81$ you have to multiply all this mess together. sorry i screwed up at the beginning.