Question

How do you write the simplest polynomial function with the zeros $2i, - 2i$ and $6$?

Answer 1

Hint: The given simplest polynomial function with the zeros$2i, - 2i$ and $6$. We find the factor that corresponds to the zero, $2i, - 2i$ and $6$. After we substitute the formula given below:
$y = $ (The factor that corresponds to the zero,$2i$) (The factor that corresponds to the zero,$2i$) (The factor that corresponds to the zero, $6$)
After that we modify in the form of $(a - b)(a + b) = {a^2} - {b^2}$
Use the property ${i^2} = - 1$
After that we simplify the equation. Finally we get the cubic equation.

Complete step by step answer:
The given simplest polynomial function with the zeros $2i, - 2i$ and $6$
The factor that corresponds to the zero,$2i$, is $(x - 2i)$.
The factor that corresponds to the zero,$ - 2i$, is $(x + 2i)$.
The factor that corresponds to the zero, $6$, is $(x - 6)$.
Collect the factors into an equation:
$y = $ (The factor that corresponds to the zero, $2i$) (The factor that corresponds to the zero, $2i$) (The factor that corresponds to the zero, $6$)
$y = (x - 2i)(x + 2i)(x - 6)$
We can use the pattern,$(a - b)(a + b) = {a^2} - {b^2}$, to multiply the first two factors:
$y = ({x^2} - 4{i^2})(x - 6)$
Use the property ${i^2} = - 1$ to simplify the first factor:
$y = ({x^2} - ( - 4))(x - 6)$
Multiply negative by negative, hence we get
$y = ({x^2} + 4)(x - 6)$
We multiply the first term by the second term
$y = {x^3} - 6{x^2} + 4x - 24$

Since, we write the simplest polynomial function with the zeros $2i, - 2i$ and $6$ are $y = {x^3} - 6{x^2} + 4x - 24$.

Note: You have learned several important properties about real roots of polynomial equations.
The following statements are equivalent:
A real number $r$ is a root of the polynomial equation $P(x) = 0$
$P(r) = 0$
$r$ is an $x$ -intercept of the graph of $P(x)$
$x - r$ is a factor of $P(x)$
When you divide the rule for $P(x)$ by $x - r$, the remainder is$0$
$r$ is a zero of $P(x)$.