posted by jordan on .
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
-2+i, -2-i, 3, -3
Zeros are also "x = ..." statements, so x = -2 + i, x = -2 - i, etc...
Since you need REAL coefficients in the polynomial, change the x = -2 + i and x = -2 - i statements to "clean" polynomials, like this:
x = -2 + i
x + 2 = i (square each side to get rid of the i)
(x+2)^2 = i^2
x^2 + 4x + 4 = -1
==> x^2 + 4x + 5 = 0
x = -2 - i
(x+2)^2 = (-i)^2
I'm not going to go further with this because you get the same exact equation as the one before. So now when you multiply all the zeroes together, you get this:
0 = (x^2 + 4x + 5)(x-3)(x+3)
Multiply it out and you get:
x^4 + 4x^3 - 4x^2 - 36x - 45 = 0
Does this make sense?
The polynomial would have to have 4 zeros, meaning it would have to be a polynomial of the 4th degree. The general form for a polynomial of the 4th degree with zeros a, b, c and d would be:
where f is a random real number (lets take this to be 1 in this case).
So, if we fill in the zeros you were given we get that:
When we multiply the first two factors and the last two, we get:
(x^2 + 2x + ix + 2x + 4 + 2i -ix - 2i +1) * (x^2 - 9) =
(x^2 + 4x + 5)*(x^2 - 9) =
(x^4 - 9x^2 + 4x^3 - 36x + 5x^2 -45) =
x^4 + 4x^3 - 4x^2 -36x - 45
this is a polynomial of the 4th degree which has the given values as its zeros