Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros ( 3i, 2 - i ). write the polynomial in standard form. ( f(x) = square )

Explanation:

Step1: Identify conjugate zeros

For rational coefficients, complex zeros come in conjugate pairs. So if $3i$ is a zero, $-3i$ is also a zero; if $2-i$ is a zero, $2+i$ is also a zero.

Step2: Write factor form

Polynomial with leading coefficient 1:
$f(x)=(x-3i)(x+3i)(x-(2-i))(x-(2+i))$

Step3: Multiply first two factors

Use $(a-b)(a+b)=a^2-b^2$:
$(x-3i)(x+3i)=x^2-(3i)^2=x^2-9i^2=x^2+9$

Step4: Multiply last two factors

First rewrite terms: $(x-2+i)(x-2-i)=((x-2)+i)((x-2)-i)$
Apply difference of squares:
$(x-2)^2-i^2=x^2-4x+4+1=x^2-4x+5$

Step5: Multiply the two quadratics

$$\begin{align*} f(x)&=(x^2+9)(x^2-4x+5)\\ &=x^2(x^2-4x+5)+9(x^2-4x+5)\\ &=x^4-4x^3+5x^2+9x^2-36x+45 \end{align*}$$

Step6: Combine like terms

Combine $5x^2$ and $9x^2$:
$f(x)=x^4-4x^3+14x^2-36x+45$

Answer:

$f(x)=x^4-4x^3+14x^2-36x+45$