Question

Graph each function using the techniques of shifting,compressing,, and or reflecting.Start with the graph of the basic function

Answer 1

we know that

The given zeros are

x=0

x=i

x=1+i

Remember that

the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P

so

the other zeros of the given polynomial f(x) are

x=-i

x=1-i

The polynomial in factored form is given by

$f(x)=x(x-i)(x+i)(x-(1+i))(x-(1-i))$

Simplify

$\begin{gathered} f(x)=x(x^2-i^2)(x^2-x(1-i)-x(1+i)+(1-i^2)) \\ f(x)=x(x^2+1)(x^2-x+xi-x-xi+1+1) \\ f(x)=x(x^2+1)(x^2-2x+2) \end{gathered}$

simplify

$\begin{gathered} f(x)=(x^3+x)(x^2-2x+2) \\ f(x)=x^5-2x^4+2x^3+x^3-2x^2+2x \\ f(x)=x^5-2x^4+3x^3-2x^2+2x \end{gathered}$