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.  


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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.


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