What are all the real and complex roots of the polynomial 3x

What are all the real and complex roots of the polynomial 3x3 + 30x2 + 159x + 522, given that one root is x = -6? To receive full credit you must simplify your answers as much as possible. Enter real numbers as either integers or decimals (not fractions). Use i for the sqrt(-1).

Solution

3x³ + 30x² + 159x + 522 = 3x³ + 18x² + !2x² + 72x + 87x + 522 = 3x²( x + 6) + 12x(x +6) + 87( x + 6) = (x + 6)( 3x² + 12x + 87). Now, the roots of 3x² + 12x + 87 are [ -12 ± { (12)²– 4(1)(87)} ]/ 2*1 i.e [ -12 ± (144 – 348)]/ 2 or, [ - 12 ± (- 204)]/2 or, (-12 ± 14.28 i ) / 2 i.e. - 6 ± 7.14 I i.e. -6 + 7.14i and – 6 – 7.14i . Thus the roots of 3x³ + 30x² + 159x + 522  are x = - 6 , x = -6 + 7.14 i and x = – 6 – 7.14i

NOTE:

1. We have rounded off to 2 decimals while computing the square root of 204.

2. We have used the formula that the roots of ax² + bx + c are x = [ -b ± (b² – 4ac)] / 2a