Question

A) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list.

Answer 1

Answer:

steps below

Step-by-step explanation:

x⁶ - 7x³ - 8 = (x⁶ + x³) - (8x³ + 8)

                  = x³ (x³+1)-8(x³+1)

                  = (x³ - 2³)(x³ + 1)

                  = (x-2)(x²+2x+4)(x+1)(x²-x+1)

(x-2)(x²+2x+4)(x+1)(x²-x+1) = 0

x= -1 or x = 2   ... roots

or    x²+2x+4=0   or    x²-x+1=0  

x²+2x+4=0

x = (-2±√4-16)/2 = -1 ± √3 i   ... complex roots

x²-x+1=0

x = (1±√1-4)/2 = 1/2 ± (√3)i / 2   ... complex roots

b) P(x) = x⁶ - 7x³ - 8

           = (x-2)(x²+2x+4)(x+1)(x²-x+1)

           = (x+1)(x-2)(x-(-1 + √3 i))(x-(-1 - √3 i)(x-(1/2 + (√3)i / 2))(x-(1/2 - (√3)i / 2))