SOLUTION: find all the zeros of the equation x^4-3x^3+6x^2+2x-60=0 given that 1+3i is a zero

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Question 827428: find all the zeros of the equation x^4-3x^3+6x^2+2x-60=0 given that 1+3i is a zero
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!

x4-3x³+6x²+2x-60 = 0

Use synthetic division:

1+3i | 1  -3      6     2      -60
     |     1+3i -11-3i  4-18i   60
       1  -2+3i  -5-3i  6-18i    0

So the first factorization is:

[x-(1+3i)][x³+(-2+3i)x²+(-5-3i)x+(6-18i)] = 0

Since 1+3i is a solution, so is its conjugate 1-3i

1-3i | 1  -2+3i  -5-3i  6-18i
     |     1-3i  -1+3i -6+18i
       1  -1     -6

The second factorization is:

[x-(1+3i)][x-(1-3i)](x²-x-6) = 0

The third and final factorization is:

[x-(1+3i)][x-(1-3i)](x-3)(x+2) = 0

x-(1+3i)=0
       x=1+3i

x-(1-3i)=0
       x=1-3i

     x-3=0
       x=3

     x+2=0
       x=-2

The four solutions are 1+3i, 1-3i, 3, -2

Edwin
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