Given: (2x - 3y^2)^9
Find: a.) sum of numerical coefficients
b.) term that includes y^10

Question

Given: (2x - 3y^2)^9
Find: a.) sum of numerical coefficients
b.) term that includes y^10

campbell_st · Answer

any thoughts on which you need help with..?

stana · Answer

i was wondering if there's a way to find the sum of the coefficients without having to solve them one by one

adrimit · Answer

use a binomial expansion. the only term with y^10 in it is $$(a_n)x^4*y^{10}$$
$$a_n=2^{4}(-3)^{5}\frac{9!}{4!5!}$$

campbell_st · Answer

well not really, but I would just write then using combination notation and index laws

maybe using summation notation 

$$\sum_{r=0}^{9} C_{r}^{n} (2)^{ n - r}~(-3)^r$$

that would be my best guess for part A

adrimit · Answer

also, the sum of the coefficients is equal to the expanded polynomial evaluated at the point (1,1)
$$p(1,1)=\left(2(1)-3(1^2)\right)^9=\left(2-3\right)^9=(-1)^9=-1$$

adrimit · Answer

another more general example is the following:
find the sum of all coefficients of (x+y-2z)^3 that contain only x, z or both
$$p(1,0,1)=(-1)^3=-1
\(x+y)^3-6z(x+y)^2+12z^2(x+y)-8z^3
\(x^3+3x^2y+3xy^2+y^3)-6z(x^2+2xy+y^2)+12z^2(x+y)-8z^3
\eliminate\;all\;terms\;that\;contain\;y
\x^3-6zx^2+12z^2x-8z^3
\1-6+12-8=13-14=-1$$