Question

Find a polynomial f(x) of degree 3 with real coefficients and the following zeros. -2,- 2+i. ​

Answer 1

Answer:

f(x) = x^3 + 2x^2 + x + 2

Step-by-step explanation:

If -2 + i is a zero, then -2 - i must also be a zero.

These zeros are -2, -2 + i and -2 - i, and the corresponding factors of the polynomail f(x) are (x + 2), (x + 2 - i) and (x + 2 + i).

(x + 2 - i) and (x + 2 + i) multiplied together becomes x^2 + 1.

Multiplying this result by the 3rd and last factor, (x + 2), we get

(x^2 + 1)(x + 2) = x^3 + 2x^2 + x + 2

Check, using synthetic division.  What are the factors of x^3 + 2x^2 + x + 2?

-2   /   1    2    1    2  

              -2   -2   -2

     -----------------------

         1      0    -1    0                  Since the remainder is zero, we know that

                                                    (x + 2) is a factor and -2 is a root.