Chapter 3, Polynomial and Rational Functions - Section 3.5 - Complex Zeros and the Fundamental Theorem of Algebra - 3.5 Exercises - Page 329: 34

The zeros of the function are: $\{0,\sqrt 7i,-\sqrt 7i\}$ The complete factorization of P is: $P(x)=x^{3}(x+\sqrt 7i)(x-\sqrt 7i)$ $x =0$ with multiplicity 3 $x =\sqrt 7i$ with multiplicity 1 $x =-\sqrt 7i$ with multiplicity 1

Factor the polynomial completely to obtain: $P(x)=x^{5}+7x^{3}$ $P(x)=x^{3}(x^{2}+7)$ $P(x)=x^{3}(x+\sqrt 7i)(x-\sqrt 7i)$ Equate each unique factor to zero then solve each equation to obtain: $x^{3}=0 \rightarrow x=0$ $x+\sqrt 7i=0 \rightarrow x=-\sqrt 7i$ $x-\sqrt 7i=0 \rightarrow x=\sqrt 7i$ The zeros of the function are: $\{0,\sqrt 7i,-\sqrt 7i\}$ The complete factorization of P is: $P(x)=x^{3}(x+-\sqrt 7i)(x-(-\sqrt 7i))$ $P(x)=x^{3}(x+\sqrt 7i)(x-\sqrt 7i)$ $x =0$ with multiplicity 3 $x =\sqrt 7i$ with multiplicity 1 $x =-\sqrt 7i$ with multiplicity 1