Question

Find the complex zeros of

`f(x) = {x}^(3 ) - 13 {x}^(2) + 59x - 87`
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Answer 1

`f(x) = 0 \\\\\implies x^3-13x^2+59x -87 =0\\\\\implies x^3 -3x^2 -10x^2 +30x +29x - 87=0\\\\\implies x^2(x-3) - 10x(x-3) + 29(x-3) =0\\\\\implies (x-3)(x^2 -10x +29) =0\\\\\implies x - 3 = 0 ~~ \text{or}~~x^2 -10x +29 = 0\\\\\implies x =3~~ \text{ or} ~~ x = (-(-10) \pm √((-10)^2 -4 \cdot 1 \cdot 29))/(2(1))\\\\\implies x =3~~ \text{ or} ~~ x = (10\pm √(-16))/(2)\\\\\implies x =3~~ \text{ or} ~~ x = (10\pm 4i )/(2)\\`

`\\\implies x =3~~ \text{ or} ~~ x =5 \pm 2i\\\\\text{Hence the complex roots are,}~ 5- 2i ~~ \text{and} ~~5 + 2i`