Question

homework assignment 3.6: zeros of polynomials
score: 2/12 answered: 2/12
question 3
find a degree 3 polynomial with real coefficients having zeros 1 and $2i$ and a lead coefficient of 1. write $p$ in expanded form.
$p(x) = $

Explanation:

Step1: Identify all polynomial zeros

Since the polynomial has real coefficients, if $2i$ is a zero, its complex conjugate $-2i$ is also a zero. The zeros are $1$, $2i$, $-2i$.

Step2: Write factored form of polynomial

Using the factor theorem, the polynomial with lead coefficient 1 is:
$P(x)=(x-1)(x-2i)(x+2i)$

Step3: Multiply complex factors first

Multiply $(x-2i)(x+2i)$ using the difference of squares:
$(x-2i)(x+2i)=x^2-(2i)^2=x^2-4i^2=x^2+4$

Step4: Multiply by remaining factor

Multiply $(x-1)$ with $x^2+4$:
$P(x)=(x-1)(x^2+4)=x(x^2+4)-1(x^2+4)=x^3+4x-x^2-4$

Step5: Rearrange into standard form

Order terms by descending degree:
$P(x)=x^3-x^2+4x-4$

Answer:

$x^3 - x^2 + 4x - 4$