Chapter 5 Polynomials and Polynomial Functions - 5.9 Write Polynomial Functions and Models - 5.9 Exercises - Quiz for Lessons 5.7-5.9 - Page 399: 6

$f(x)=x^5-7x^4+13x^3-31x^2+36x-12$

Using the Factor Theorem, if $x=a$ is a factor, then $(x-a)$ is a factor of $f(x)$. Hence here $f(x)=(x-(-2i))(x-1)(x-2i)(x-(3-\sqrt 6))(x-(3+\sqrt 6))\\=(x-1)(x+2i)(x-2i)(x-3+\sqrt 6)(x-3-\sqrt 6)\\=(x^2-6x+3)(x-1)(x+2i)(x-2i)\\=(x-1)(x^2+4)(x^2-6x+3)\\=(x^3-x^2+4x-4)(x^2-6x+3)\\=x^5-7x^4+13x^3-31x^2+36x-12$