Answer to Question #131598 in Calculus for Jackson

Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
8, -14, and 3 + 9i

Answer 1

According to the Fundamental theorem of algebra, every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

Therefore, the degree of our polynomial will have a degree greater or equal to 3.Moreover, our polynomial could be represented in form of $(x-x_1)(x-x_2)(x-x_3)$ if all the roots are real. But the third root is a complex number, so it is a complex root of an $x^2-ax+b$ with $a^2-4b<0$ . We know that the second root of such an equation will have form 3-9i, so the polynomial in question has to have 4 roots and degree 4. The form of the polynomial is $(x-8)(x+14)(x-(3+9i))(x-(3-9i)) = x^4-58x^2+1212x-10080$