Find all the zeros of the polynomial

Question

Find all the zeros of the polynomial P(x)=x⁶-64

its zeros are 

x1=, x2= with x1<x2

x3= ()+()i with both negative real and imaginary parts, 

x4= ()+()i with negative real part and positive imaginary part

x5= ()+()i with positive real part and negative imaginary part

x6= ()+()i with both positive real and imaginary parts. 

Answer 1

Hi Dalia,

 

P(x)=x^6-64

By factirizing, we get,

      =(x³)² - (8)²

      =(x³-8)(x³+8)........using formula x²-y²=(x-y)(x+y)

      =(x-2)(x²+2x+4)(x+2)(x²-2x+4).............................using formula x³-y³=(x-y)(x²+xy+y²)

x-2=0  , x+2=0

x=2     , x=-2

Solving x²-2x+4=0 to find complex zeros

Using formula,for ax²+bx+c=0,

   x=-b±√b²-4ac⁄2a

     substituting a=1, b=-2,c=4

we get,

x=-(-2)±√(-2)²-4(1)(4)⁄2(1)

  =2±√4-16⁄2

   =2±√-12⁄2

    =2+2√3i⁄2     or = 2-2√3 i⁄2........ using √-1= i

 x=1+√3 i   or x=1-√3 i

similarly solving for x²+2x+4=0

x=-2±2√3 i⁄2

x=- 1±√3 i

so x= -1+√3 i   or x=-1-√3 i

 

Ans: zeros are x1=-2

                      x2=2

                      x3= -1-√3 i

                      x4=-1+√3 i

                      x5=1-√3 i

                      x6=1+√3 i

 

Meena from Strongsville, OH