Question

find a degree 3 polynomial with real coefficients having zeros 2 and 3i and a lead coefficient of 1. write p in expanded form.
p(x) =
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Explanation:

Step1: Identify all zeros

For real coefficients, complex zeros come in conjugate pairs. If $3i$ is a zero, then $-3i$ is also a zero. The zeros are $2$, $3i$, $-3i$.

Step2: Write factored form

With leading coefficient 1, the polynomial is:
$P(x)=(x-2)(x-3i)(x+3i)$

Step3: Multiply complex factors

First multiply $(x-3i)(x+3i)$ using $(a-b)(a+b)=a^2-b^2$:
$(x-3i)(x+3i)=x^2-(3i)^2=x^2-9i^2=x^2+9$ (since $i^2=-1$)

Step4: Multiply by linear factor

Multiply $(x-2)$ with $x^2+9$:
$P(x)=(x-2)(x^2+9)=x(x^2+9)-2(x^2+9)$

Step5: Expand and simplify

$P(x)=x^3+9x-2x^2-18=x^3-2x^2+9x-18$

Answer:

$x^3-2x^2+9x-18$