Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 3, -13, and 5 + 4i

Answer 1

Complex zeros exist in conjugate pairs so the fourth zero is 5 - 4i
So we have

P(x)  = ( x - 3)(x + 13)(x - (5 + 4i))(x - (5 - 4i))

   = (x - 3)(x + 13)( x - 5 - 4i)(x - 5 + 4i)

  = (x^2 + 10x - 39)(x^2 - 5x + 4ix - 5x + 25 -20i - 4ix  + 20i -16i^2)

 = (x^2 + 10x - 39)(x^2 - 10x + 41)

= x^4 - 10x^3 + 41x^2 + 10x^3  - 100x^2 + 410x  - 39x^2 + 390x - 1599

=  x^4 - 98x^2 + 800x - 1599  Answer