How do you write a 4th degree polynomial function?

Question

The function has the given zeros of 2, -2 and 1-3i.

Answer 1

Taja,

 

First, you only gave 3 roots for a 4th degree polynomial.  The missing one is probably imaginary also, (1 +3i).

 

For any root or zero of a polynomial, the relation  (x - root) = 0 must hold by definition of a root: where the polynomial crosses zero.  So for your set of given zeros, write:

 

(x - 2) = 0

(x + 2) = 0

(x - 1 + 3i) = 0

(x - 1 -  3i) = 0

 

Now just multiply all the left factors together to still equal zero because the right side is zero in each.

 

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

 

You can leave it in this form which explicitly highlights all the zeros of the 4th order polynomial, or multiply the factors out to put it into standard polynomial form with descending powers.

 

(x² - 4) (x² - 2x + 10) = 0  as an interim step reducing 4 factors to 2

 

x⁴ - 2x³ + 6x² + 8x - 40 = 0   is your 4th order polynomial in standard form that has the above zeros.