Question 934104: Find the constant term in the expansion of (-3x^5 + 4/x)^36.
Please help me solve this, thanks.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the method is shown in the following video.
https://www.youtube.com/watch?v=dANFpZ95kSs
basically, you solve the exponent that allows x to be equal to 1.
this occurs when the combined exponent of x is equal to 0.
to make a long story short, this occurs in the 30th term of the expansion.
here's why.
the binomial expansion formula is for (a + b)^n
the formula for the r term in the binomial expansion is c(n,r) * a^(n-r) * b^r.
our equation is (-3x^5 + 4/x)^36
in our equation, the formula for the r term in the expansion becomes:
r term = c(36,r) * (-3x^5)^(36-r) * (4/x)^r
4/x can be written as 4x^(-1), so our formula for the r term in our expansion becomes:
r term = c(36,r) * (-3x^5)^(36-r) * (4x^(-1))^r
by using the rules of exponent arithmetic, we can simplify this equation.
the rules of exponents we will be using are:
(a*b)^c = a^c * b^c
(a^b)^c = a^(b*c)
a^b * a^c = a^(b+c)
using these rules, we simplify our equation as follows:
r term = c(36,r) * (-3x^5)^(36-r) * (4x^(-1))^r becomes:
r term = c(36,r) * (-3)^(36-r) * (x^5)^(36-r) * 4^r * (x^(-1))^r which becomes:
r term = c(36,r) * (-3)^(36-r) * x^(5*(36-r) * 4^r * x^(-1*r) which becomes:
r term = c(36,r) * (-3)^(36-r) * x^(180-5r) * 4^r * x^(-r) which becomes:
r term = c(36,r) * (-3)^(36-r) * 4^r * x^(180-6r).
we need the exponent of x to be equal to 0 so we set 180 - 6r equal to 0 and solve for r to get r = 30.
the constant term in the expression will be when r = 30.
to see what the value of the constant term is, we need to evaluate the r term when r = 30.
we get r term = c(36,30) * (-3)^(36-30) * 4^30 * x^(180 - 180) which becomes:
r term = c(36,30) * (-3)^6 * 4^30 * x^0 which becomes:
r term = c(36,30) * (-3)^6 * 4^30
c(36,30) is the combination formula of 36! / (6! * 30!) which is equal to 1947792.
r term becomes 1947792 * (-3)^6 * 4^30).
use your calculator to evaluate to get:
r term = 1.637079786 * 10^27
that's your constant term.
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