Question

Help to find the real and complex zeroes of:
(x - 3)(x + 2)(x+2i)(x - 2i)

Answer 1

you can set the individual factors (x-3) , (x+2), etc . equal to zero

Answer 2

those will give you the 'zeroes' . this is a term i dislike, i prefer to call them roots or solutions

Answer 3

Well, based on the zero product property, when you have for example,
\(\large\color{black}{ (x-\color{blue}{a})(x \color{red}{+ b} )(x-\color{green}{c} )}\)
then you know when,
\(\large\color{black}{ x=\color{blue}{a},~~ \color{red}{- b}, ~~~~~OR~~~~~\color{green}{c} }\) then the expression is zero, right?

Answer 4

Yup.
The original question states:
Is it possible to create a fourth-degree polynomial with only 2 real zeros? Demonstrate how to do this and explain your steps.

Answer 5

Whatever makes one of the parenthesis, --- one part of the product, to be zero.
And you just need to solve for x in your head in 4 cases:
\(\large\color{black}{ 1)~~x - 3=0}\)
\(\large\color{red}{ 2)~~x + 2=0}\)
\(\large\color{black}{ 3)~~x+2i=0}\)
\(\large\color{red}{ 4)~~x - 2i=0}\)

Answer 6

And for your original question, you will have a positive answer. Want a quick explanation why?

Answer 7

I mena yes, by a positive answer.

Answer 8

So, you can have this,
\(\large\color{black}{ \rm (quadratic~~~~~polynomial) \color{red}{(quadratic~~~~~polynomial)=0}}\)
where the black has real solutions, and red has imaginary solutions.

Answer 9

The impossible part would be to have 3 imaginary and 1 real, or 1 imaginary and 3 reals.

Answer 10

The reason for this is, because for at least 1 polynomial out of 2, you will need one imaginary and one real solutions, and you know that in a quadratic polynomial, EITHER;
1) both solutions are real
2) both solutions are imaginary

Answer 11

If anything doesn't make sense, I"ll elaborate...;0

Answer 12

I think i understand :)
Is that it?

Answer 13

yes:)

Answer 14

As long as you don't have a situation where you get a quadratic polynomial with 1 imaginary solution, and one real.

again, either both solutions are real, or they are both imaginary.
(whether they are extraneous or not).