I need help with "Find a polynomial of lowest degree
with interger coefficients that has the indicated zeroes:
1+i,1-i,1,3"

Rule:
For a polynomial to have a zero of r, it must have 
(x - r) as a factor.

Therefore,
For the desired polynomial to have a zero of 1+i, it must have 
[x - (1+i)] as a factor.

For the desired polynomial to have a zero of 1-i, it must have 
[x - (1-i)] as a factor.

For the desired polynomial to have a zero of 1, it must have 
(x - 1) as a factor.

For the desired polynomial to have a zero of 3, it must have 
(x - 3) as a factor.

Therefore the desired polynomial must be equivalent to:

[x - (1+i)][x - (1-i)](x - 1)(x - 3)

Removing the inner parentheses in the first two factors:

[x - 1 - i][x - 1 + i](x - 1)(x - 3)

Grouping the (x - 1) in parentheses:

[(x - 1) - i][(x - 1) + i](x - 1)(x - 3)

Using FOIL on the first two factors:

[(x-1)2 + i(x - 1) - i(x - 1) - i2](x - 1)(x - 3)

The two middle terms in the brackets, i(x - 1) and -i(x - 1)
cancel, and i2 = -1

[(x-1)2 - (-1)](x - 1)(x - 3)
 
[(x-1)(x-1) + 1)](x - 1)(x - 3)

[x2-2x+1 + 1](x - 1)(x - 3)

[x2 - 2x + 2](x - 1)(x - 3)

Use FOIL on the last two factors

[x2 - 2x + 2](x2 - 4x + 3)

Group the first two terms in each factor:

[(x2-2x) + 2][(x2-4x) + 3]

Use FOIL

(x2-2x)(x2-4x) + 3(x2-2x) + 2(x2-4x) - 6

Use FOIL on first term, remove parentheses
in 2nd and 3rd terms

x4 - 6x3 + 8x3 + 3x2 - 6x + 2x2 - 8x + 6

x4 - 6x3 + 13x2 - 14x + 6  

Edwin