Question

Question #832b7

Answer 1

`= 360*a^7*b*c^2+840*a^6*b^3*c+252*a^5*b^5`

Explanation:

`"Name"`
`p(x) = b*x+c*x^2 = x(b+c*x)`
`"Then we have"`
`(a+p(x))^10 = sum_{i=0}^{i=10} C(10,i)* a^(10-i)* p(x)^i`
`= sum_{i=0}^{i=10} C(10,i)* a^(10-i)* x^i*(b+c*x)^i`
`"with "C(n,k) = (n!)/((n-k)!k!) " (combinations)"`

`= sum_{i=0}^{i=10} C(10,i)* a^(10-i)* x^i*[sum_{j=0}^{j=i} C(i,j)* b^(i-j)*(c*x)^j]`

`"coefficient of "x^5" means that "i+j =5 => j = 5-i"."`
`=> C5 = sum_{i=0}^{i=5} C(10,i)*C(i,5-i)*a^(10-i)*b^(2*i-5)*c^(5-i)`
`=> C5 = C(10,3)*C(3,2)*a^7*b*c^2+C(10,4)*C(4,1)*a^6*b^3*c+C(10,5)*C(5,0)*a^5*b^5`
`= 360*a^7*b*c^2+840*a^6*b^3*c+252*a^5*b^5`