find the complex zeros of the polynomial function.

Question

please help me with this!.

 

find the complex zeros of the polynomial function. write f in factored form.  use complex zeros to write f in factored form f(x)= x^3-6x^2+21x-26

Answer 1

f(x)=x³-6x²+21x-26. Let's see if any rational zeroes are actual zeroes.

Factors of 26 are 1, 2, 13, 26. We consider positive and negative versions of these numbers.

f(1)=1-6+21-26=-10, so x=1 is not a zero; f(-1)=-1-6-21-26=-54, so x=-1 is not a zero.

f(2)=8-24+42-26=0, so x=2 is a zero and x-2 is a factor.

Synthetic division by this zero gives us:

2 | 1 -6 21 | -26

     1  2 -8 |   26

     1 -4 13 |   0 = x²-4x+13 has complex zeroes:

Let x²-4x=-13, x²-4x+4=-9, (x-2)²=-9, x-2=±3i, so x=2±3i.

f(x)=(x-2)(x-2-3i)(x-2+3i).