# Finding Binomial Co-efficient from pronumerals

by ukee1593
Tags: binomial coeff, expansions, powers
 P: 2 1. The problem statement, all variables and given/known data I'm asked to find (a/b) in the simplest form if the co-efficient of x^8 is zero in the expansion of: (1 + x)(a - bx)^12 2. Relevant equations Binomial expansion formula ... (a + b)^n = Sum of r --> n (r = 0) (nCr)(a^(n-r) * b^r 3. The attempt at a solution I figured that x^8 could be achieved from two possible situations ... either 1 * the expansion of (a - bx)^12 or x * the expansion of (a - bx)^12 I found the value of r at both these points by looking for the value of r that makes bx^r = x^8 and x*bx^r = x^8. This I found to be r = 7 and r = 8. Then I wrote as: (12C7) * (a)^5 * (-b)^7 + (12C7) * (a)^4 * (-b)^8 = 0 Then I get stuck as I cannot seem to get values for a and b from this. Can anyone help me?
 P: 2 Problem solved ... First of all, note my error in the last line (12C7) * (a)^5 * (-b)^7 + (12C7) * (a)^4 * (-b)^8 = 0 Should be: (12C7) * (a)^5 * (-b)^7 + (12C8) * (a)^4 * (-b)^8 = 0 The negative b on the (-b)^7 comes out the front as it is an odd power ... The other negative cancels out -(12C7) * (a)^5 * (b)^7 + (12C8) * (a)^4 * (b)^8 = 0 throw the negative section over to the other side of the equals sign ... (12C8) * (a)^4 * (b)^8 = (12C7) * (a)^5 * (-b)^7 then evaluate => (495)*(a^4)*(b^8) = (792)*(a^5)*(b^7) start cancelling => (495)*(b) = (792)*(a) divide to each side => 495/792 = a/b => a/b = 5/8 which is correct according to the answers.

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