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# Need help

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I think I need to use the Binomial Theorem here, but I don't know how.

Find the coefficient of y^4 in the expansion of (2y - 5 + y^2)^6.

Apr 6, 2023

#1
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The coefficient of y^4 is -115.

Apr 6, 2023
#2
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expand  (2y - 5 + y^2)^6=

y^12 + 12 y^11 + 30 y^10 - 140 y^9 - 585 y^8 + 792 y^7 + 4164 y^6 - 3960 y^5 - 14625 y^4 + 17500 y^3 + 18750 y^2 - 37500 y + 15625

y^4 = -14625

Apr 6, 2023
#3
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Thanks guest, I am just going to look at it a different way :)

Find the coefficient of y^4 in the expansion of (2y - 5 + y^2)^6.

\([(2y - 5) + y^2]^6\)

the general term will be      \(\binom{6}{r}(y^2)^r(2y-5)^{6-r}\)

We are looking for the y^4 term so r can only be 0,1, or 2  (becasue after that I can see the y will have at least a power of 6

you need to do the three seperately and add the x^4 terms together to get the final answer.

I will do r =1,   that is the hardest one anyway.

\(\binom{6}{r}(y^2)^r(2y-5)^{6-r}\)

\(\binom{6}{1}(y^2)(2y-5)^{5}\)

So now I need to find the coefficient of  y^4   in   \((2y-5)^{5}\)

That will be

\(\binom{5}{2}(2y)^2(-5)^3\\ =10*4*-125y^2\\ =-5000y^2\)

so the y^4 term will be

\(\binom{6}{1}(y^2)(-5000y^2)\\ =-30000y^4\)

Now you have to do y=0 and y=2   and add them all together.

Apr 8, 2023
edited by Melody  Apr 8, 2023