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anonymous

  • 5 years ago

find the coefficient of x^4 in the expansion of (3x-1)^11

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  1. anonymous
    • 5 years ago
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    same as before \[\dbinom{11}{4}(3x)^4(-1)^7\]

  2. anonymous
    • 5 years ago
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    and what does the 11 over 4 part mean again?

  3. anonymous
    • 5 years ago
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    \[3^4=81\] \[(-1)^7=-1\]

  4. anonymous
    • 5 years ago
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    do you know how to find \[\dbinom{11}{4}\]? if not i will be happy to show you.

  5. anonymous
    • 5 years ago
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    it is called "eleven choose 4" the number of ways to choose 4 items out of 11.

  6. anonymous
    • 5 years ago
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    show please!! also why did u just do 3^4 isnt ther an x in there like 3x^4?

  7. anonymous
    • 5 years ago
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    \[(3x)^4=3^4x^4\] and since you were asked for the coefficient that is the number you need to compute.

  8. anonymous
    • 5 years ago
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    it is the whole thing raised to the power of 4, not just the x. that is it is \[(3x)^4\] not \[3x^4\]

  9. anonymous
    • 5 years ago
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    so there for it is -81x^4

  10. anonymous
    • 5 years ago
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    if i times it by the -1 like i was supposed to?

  11. anonymous
    • 5 years ago
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    now to compute \[\dbinom{11}{4}\] yes there is a -81

  12. anonymous
    • 5 years ago
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    but you have to multiply by \[\dbinom{11}{4}\] which is easy enough to compute. i can show you step by step if you like

  13. anonymous
    • 5 years ago
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    thatd be great!

  14. anonymous
    • 5 years ago
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    ok first of all a formula, although you don't really use it. the formula is \[\dbinom{n}{k}=\frac{n!}{k! (n-k)!}\] here n = 11, k = 4 and n-k=7 make a fraction. in the numerator put 4 numbers starting at 11 and counting down. in the denominator you put 4! to get \[\dbinom{11}{4}=\frac{11\times 10 \times 9\times 8}{4 \times 3\times 2}\]

  15. anonymous
    • 5 years ago
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    is my final answer -26730x^4?

  16. anonymous
    • 5 years ago
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    now since this is a whole number , cancel first and multiply last! \[\frac{11 \times 10\times 9 \times 8}{4\times 3\times 2}={11\times 10\times 3}\]

  17. anonymous
    • 5 years ago
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    ?

  18. anonymous
    • 5 years ago
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    \[\dbinom{11}{4}=330\] \[330\times 81 \times -1=-26730\] yes you got it!

  19. anonymous
    • 5 years ago
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    thankyou so much!!!!!!! your awesome!

  20. anonymous
    • 5 years ago
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    can i ask u about another??!

  21. anonymous
    • 5 years ago
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    no problem. try \[\dbinom{10}{3}\] and convince yourself that it is the same as \[\dbinom{10}{7}\]because 3+7=10!

  22. anonymous
    • 5 years ago
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    sure ask away.

  23. anonymous
    • 5 years ago
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    find the coefficient of x^7 in the expansion of (2x-5)^9

  24. anonymous
    • 5 years ago
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    ok same idea. here n = 9, k = 7, n-k=3 so the term with \[x^7\] will look like \[\dbinom{9}{7}(2x)^7(-5)^3\]

  25. anonymous
    • 5 years ago
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    oops typo sorry

  26. anonymous
    • 5 years ago
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    n-k=2!

  27. anonymous
    • 5 years ago
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    my mistake. it will be \[\dbinom{9}{7}(2x)^7(-5)^2\]

  28. anonymous
    • 5 years ago
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    exponents have to add up to 9. would you like to try it?

  29. anonymous
    • 5 years ago
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    so what would (2x)^7 be?

  30. anonymous
    • 5 years ago
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    \[(2x)^7=2^7x^7\] so you will need to compute \[2^7\]

  31. anonymous
    • 5 years ago
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    thats is easy, as is \[(-5)^2\]

  32. anonymous
    • 5 years ago
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    your real job is to compute \[\dbinom{9}{7}\]

  33. anonymous
    • 5 years ago
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    yes how do u do the 9 over 7 thging again?

  34. anonymous
    • 5 years ago
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    \[\dbinom{n}{k}=\frac{n!}{k! (n-k)!}\]

  35. anonymous
    • 5 years ago
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    here n = 9, k = 7 and n-k=2

  36. anonymous
    • 5 years ago
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    so 9! over 7! times 2!

  37. anonymous
    • 5 years ago
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    right. but don't forget to cancel away first because the entire denominator will cancel

  38. anonymous
    • 5 years ago
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    the answer is 36?

  39. anonymous
    • 5 years ago
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    exactly!

  40. anonymous
    • 5 years ago
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    so 36 times 128 times 25

  41. anonymous
    • 5 years ago
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    so 25200x^7?

  42. anonymous
    • 5 years ago
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    now i show you the easy way. first of all 7+2=9 so it is easier to compute \[\dbinom{9}{2}\] so we work as before. make a fraction. in the numerator put to numbers starting at 9 in the denominator put 2. we get the answer right away. \[\dbinom{9}{2}=\frac{9\times 8}{2}=9\times 4=36\]

  43. anonymous
    • 5 years ago
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    good!!

  44. anonymous
    • 5 years ago
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    yes, 36 times 128 times 25 is it.

  45. anonymous
    • 5 years ago
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    oh wait i got a different number than you. i got 115200

  46. anonymous
    • 5 years ago
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    maybe i put it in wrong.

  47. anonymous
    • 5 years ago
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    no i think i am right.

  48. anonymous
    • 5 years ago
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    nah i put it in wrong!

  49. anonymous
    • 5 years ago
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    whew i was scared but it is late.

  50. anonymous
    • 5 years ago
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    is that enough of this? or are there more?

  51. anonymous
    • 5 years ago
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    you have to go?

  52. anonymous
    • 5 years ago
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    i can help you with another if you like.

  53. anonymous
    • 5 years ago
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    one question quick.. the fianl answer is 115200x^7 rights?

  54. anonymous
    • 5 years ago
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    final answer yes.

  55. anonymous
    • 5 years ago
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    nah its okay i will let u go! u were a great help! quick question .. ur suyper smart!! how old are u?!

  56. anonymous
    • 5 years ago
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    old as black pepper. have fun, and don't forget to convince yourself that \[\dbinom{10}{6}=\dbinom{10}{4}\]because 10=4+6

  57. anonymous
    • 5 years ago
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    awesome thanks!! haha

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