College Algebra, Chapter 4, 4.5, Section 4.5, Problem 72

a.) Determine the polynomial with real coefficients of the smallest possible degree for which $i$ and $1 + i$ are zeros and in which the coefficient of the highest power is 1.
b.) Determine the polynomial with complex coefficients of the smallest possible degree for which $i$ and $1 + i$ are zeros and in which the coefficient of the highest power is 1.

a.) Recall that if the polynomial function $P$ has real coefficients and if $a + b i$ is a zero of $P$, then $a - bi$ is also a zero of $P$.
In our case, the zeros of $P$ are $i, -i, 1 + i$ and $1 - i$. Thus, the required polynomial has the form
$P(x) = a(x -i)(x+i) [ x - (1 +i)] [ x-(1-i)]$ Model

$\begin{equation} \begin{aligned} P(x) &= a ( x -i )(x + i) [ (x -1) - i][ (x -1 ) + i] && \text{Regroup}\\ \\ &= a(x^2 - i^2) \left[(x-1)^2 - i^2 \right] && \text{Difference of squares}\\ \\ &= a(x^2 - i^2) \left[ x^2 - 2x + 1 - i^2\right] && \text{Expand}\\ \\ &= a(x^2 +1) \left[ x^2 - 2x + 1 + 1 \right] && \text{Recall that } i^2 = -1\\ \\ &= a(x^2 + 1) \left(x^2 - 2x + 2 \right) && \text{Add the constants}\\ \\ &= a \left[ x^4 - 2x^3 + 2x^2 + x^2 - 2x + 2 \right] && \text{Expand}\\ \\ &= a \left[ x^4 - 2x^3 + 3x^2 - 2x + 2 \right] && \text{If the coefficient of the highest power is 1, then } a = 1\\ \\ &= x^4 - 2x^3 + 3x^2 - 2x + 2 \end{aligned} \end{equation}$


b.) If $i$ and $1 + i$ are zeros, then the request polynomials has the form

$\begin{equation} \begin{aligned} P(x) &= a(x-i)[x - (1 + i)] && \text{Model}\\ \\ &= a(x-i)(x-1-i) && \text{Distribute the negative sign}\\ \\ &= a\left( x^2 - x - ix - ix + i - i^2 \right)&& \text{Expand}\\ \\ &= a \left( x^2 - x - 2ix + i - i^2 \right) && \text{Combine like terms}\\ \\ &= a \left( x^2 - (1 + 2i) x + i + 1 \right) && \text{Simplify, recall that } i^2 = -1\\ \\ &= x^2 - (1 + 2i)x + i + 1 && \text{If the coefficient of the highest power is 1, then } a = 1 \end{aligned} \end{equation}$