Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

5, -3, and -1 + 2i

Answer 1

x^4 - 15x^2 - 38x - 60.

Explanation:

Writing it in factor form:

( x - 5)(x + 3) ( x - (-1 + 2i) )(x - (-1 - 2i))

There are 4 parentheses because complex roots occur in pairs.

( x - (-1 + 2i) )(x - (-1 - 2i))

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

= x^2 + x + 2ix + x + 1 + 2i - 2ix - 2i + 4

= x^2 + 2x + 4.

So our polynomial is

( x - 5)(x + 3)( x^2 + 2x + 4)

= (x^2 - 2x - 15)(x^2 + 2x + 4)

= x^4 + 2x^3 + 4x^2 - 2x^3 - 4x^2 - 8x - 15x^2 - 30x - 60

= x^4 - 15x^2 - 38x - 60.