Question

15 Points!

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

A) f(x) = x^4 - 4x^3 + 10x^2 + 20x + 75

B) f(x) = x^4 - 14x^2 - 40x - 75

C) f(x) = x^4 - 4x^3 - 10x^2 - 20x - 75

D) f(x) = x^4 + 10x^2 - 40x - 75

Answer 1

`5; -3; -1 -2i`
In polynomials we have not of an imaginary unit (i), therefore we have the fourth zeros -1-2i.


`f(x)=(x-5)(x+3)\left[x-(-1-2i)\right]\left[x-(-1+2i)\right]\\\\=(x^2+3x-5x-15)[x^2-x(-1+2i)-x(-1-2i)+(-1)^2-(2i)^2]\\\\=(x^2-2x-15)(x^2+x-2i+x+2i+1+4)\\\\=(x^2-2x-15)(x^2+2x+5)\\\\=x^4+2x^3+5x^2-2x^3-4x^2-10x-15x^2-30x-75\\\\=x^4-14x^2-40x-75`

Answer: B)