Question 60256
<pre>I need help with "Find a polynomial of lowest degree
with interger coefficients that has the indicated zeroes:
1+i,1-i,1,3"
<font size = 4 color = "darkgreen"><b>
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)<sup>2</sup> + i(x - 1) - i(x - 1) - i<sup>2</sup>](x - 1)(x - 3)

The two middle terms in the brackets, i(x - 1) and -i(x - 1)
cancel, and i<sup>2</sup> = -1

[(x-1)<sup>2</sup> - (-1)](x - 1)(x - 3)
 
[(x-1)(x-1) + 1)](x - 1)(x - 3)

[x<sup>2</sup>-2x+1 + 1](x - 1)(x - 3)

[x<sup>2</sup> - 2x + 2](x - 1)(x - 3)

Use FOIL on the last two factors

[x<sup>2</sup> - 2x + 2](x<sup>2</sup> - 4x + 3)

Group the first two terms in each factor:

[(x<sup>2</sup>-2x) + 2][(x<sup>2</sup>-4x) + 3]

Use FOIL

(x<sup>2</sup>-2x)(x<sup>2</sup>-4x) + 3(x<sup>2</sup>-2x) + 2(x<sup>2</sup>-4x) - 6

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

x<sup>4</sup> - 6x<sup>3</sup> + 8x<sup>3</sup> + 3x<sup>2</sup> - 6x + 2x<sup>2</sup> - 8x + 6

x<sup>4</sup> - 6x<sup>3</sup> + 13x<sup>2</sup> - 14x + 6  

Edwin</pre>