Chapter 3, Polynomial and Rational Functions - Section 3.5 - Complex Zeros and the Fundamental Theorem of Algebra - 3.5 Exercises - Page 329: 33

The zeros of the function are: $\{1,-1,2i,-2i\}$ The complete factorization of P is: $P(x)=(x-2i)(x+2i)(x-1)(x+1)$ $x =-2i$ with multiplicity 1 $x =2i$ with multiplicity 1 $x =1$ with multiplicity 1 $x =-1$ with multiplicity 1

Factor the polynomial completely to obtain: $P(x)=x^{4}+3x^{2}-4$ $P(x)=(x^{4}+4x^{2})-(x^{2}+4)$ $P(x) = x^{2}(x^{2}+4)-(x^{2}+4)$ $P(x)=(x^{2}+4)(x^{2}-1)$ $P(x)=(x-2i)(x+2i)(x-1)(x+1)$ Equate each unique factor to zero then solve each equation to obtain: $x-2i=0 \rightarrow x=2i$ $x+2i=0 \rightarrow x=-2i$ $x+1=0 \rightarrow x=-1$ $x-1=0 \rightarrow x=1$ The zeros of the function are: $\{1,-1,2i,-2i\}$ The complete factorization of P is: $P(x) = (x-2i)(x-(-2i))(x-1)(x-(-1))$ $P(x)=(x-2i)(x+2i)(x-1)(x+1)$ $x =-2i$ with multiplicity 1 $x =2i$ with multiplicity 1 $x =1$ with multiplicity 1 $x =-1$ with multiplicity 1