Question

Given that a + b(1+x)^3 + c(1+2x)^3 + d(1+3x)^3 = x^3,

find the values for a, b c, d.

Answer 1

so do i need to expand the things using binomial, or what??

Answer 2

because it's a mixture of AP's and GP's.

Answer 3

use binomial please if confused ask me please

Answer 4

so i expanded it, but everything is in b's and c's and d's.

Answer 5

so use systems of equations.

Answer 6

but then everything is equated to 0.

Answer 7

does anyone have any idea as to what i could do here??

Answer 8

x^3+d (-1-9 x-27 x^2-27 x^3)+b (-1-3 x-3 x^2-x^3)-c(1+2 x)^3=a

Answer 9

a + b(1+x)^3 + c(1+2x)^3 + d(1+3x)^3 = x^3

a + b(1 + 3x + 3x^2 + x^3) + c(1 + 6x + 12x^2 + 8x^3) + d(1 + 9x + 27x^2 + 27x^3)=x^3

Now leave the RHS, and group the LHS accordingly:

(b + 8c + 27c)x^3 + (3b + 12c + 27d)x^2 + (3b + 6c + 9c)x + (a + b + c + d) = x^3

it follows that:

b + 8c + 27c=1
3b + 12c + 27d=0
3b + 6c + 9c=0
a + b + c + d =0

solve that.