Question

The coefficient of x^3 in the expansion of (2+x)(3-ax)^4 is 30. Find the values of the constant a.

Answer 1

1. (3 - ax)^4 = (ax - 3)^4 = (ax)^4 - 4(ax)^3(3) + 6(ax)^2(3^2) - 4(ax)(3^3) + 3^4

now the resulting cubic term when we multiply the above with (2+x) ... comes from
2*(-4)(ax)^3(3) + x(6(ax)^2(3^2))

coefficient...
-24a^3 + 54a^2 = 30 .... divide by -6

4a^3 - 9a^2 = -5
then one solution is a = 1.

continuing...

4a^3 - 9a^2 + 5 = 0 .. has factor a-1
(4a^3 - 9a^2 + 5)/(a-1) = 4a^2 - 5a - 5 = 0

the other roots can be found by quadratic formula...

Answer 2

thanks