a) Two of the roots of the equation x^3 +3x^2 -4x+d=0 are opposites. Find the value of d and the three roots.
a) We want the roots of `x^3 +3x^2 -4x + d=0`
If we are given that two of the roots are opposites then we have that
`x^3 + 3x^2 - 4x + d = (x-a)(x+a)(x-b)`
`= (x^2 -a^2)(x-b)`
Multiply this out
`(x^2-a^2)(x-b) = x^3 - bx^2 - a^2x + a^2b`
Equating terms we get ` `` ``a=2`, `b=-3` and `d=-12`.
Check: `(x-2)(x+2)(x+3) = (x^2-4)(x+3) `
`= x^3 + 3x^2 - 4x - 12`
The roots are +/- 2 and -3.