Chapter 5 Polynomials and Polynomial Functions - 5.7 Apply the Fundamental Theorem of Algebra - 5.7 Exercises - Skill Practice - Page 385: 57

See below

Given: $f(x)=x^3+2x^2+2i-2;-1+i$ Apply the property: $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ For $f(-1+i)\\=[(-1)^3+3(-1)^2(i)+3(-1)(i)^2+(i)^3]-2[(-1)^2+2(-1)(i)+(i)^2]+2(-1+i)-2\\=2+2i-4i+2i-2\\=0$ Thus, $-1+i$ is a zero of the given polynomial. For $f(-1-i)\\=[(-1)^3+3(-1)^2(-i)+3(-1)(-i)^2+(-i)^3]-2[(-1)^2+2(-1)(-i)+(-i)^2]+2(-1-i)-2\\=2-2i+4i+2i-2\\=4i$ Thus, $-1-i$ is not a zero of the given polynomial.