Question

16 multiple choice 1 point

1. write a polynomial function of least degree that has zeros 2 and 3i.

a. $x^2 + x - 6 = 0$
b. $x^2 - 5x - 6 = 0$
c. $x^3 - 2x^2 - 9x + 18 = 0$
d. $x^3 - 2x^2 + 9x - 18 = 0$
a b c d

Explanation:

Step1: Identify all roots

For polynomials with real coefficients, complex roots come in conjugate pairs. If $3i$ is a root, then $-3i$ is also a root. The given roots are $2$, $3i$, and $-3i$.

Step2: Form linear factors

Convert each root to a linear factor:
$(x-2)$, $(x-3i)$, $(x+3i)$

Step3: Multiply complex factors first

$$(x-3i)(x+3i) = x^2 - (3i)^2 = x^2 - 9i^2 = x^2 + 9$$

Step4: Multiply with remaining factor

$$(x-2)(x^2 + 9) = x(x^2+9) - 2(x^2+9) = x^3 +9x -2x^2 -18$$
Rearrange terms: $x^3 -2x^2 +9x -18=0$

Answer:

D. $x^3 - 2x^2 + 9x - 18 = 0$