Imaginary Zeros of a polynomial function

[xxMsJojoxx]

I'm not fully understanding the meaning of the 'imaginary zero' of this function.
I understand the '-3' as a zero, because this is clearly when y=0. But where is the imaginary zero of on the graph? I don't' understand what is the significance of this imaginary zero, and what it means.

 

[Otis]

… where is the imaginary zero of

on the graph?

Hi Ms Jojo. We don't see imaginary zeros on xy-coordinate graphs because they are not Real numbers. Such graphs show only Real x-values and Real y-values.

You may confirm the imaginary roots by substituting them (one at a time) for symbol x in f(x) and then evaluating to obtain zero.

I don't' understand what is the significance of this imaginary zero, and what it means …

That's a good research question! A search engine will lead you to some answers. Mathematicians and computer scientists are often interested in function zeros, whether they're Real or not. (The imaginary unit was developed so that mathematicians could find solutions to equations like x^2+1=0.) In physics and engineering, a general example of their significance is that roots containing an Imaginary component serve to indicate specific behaviors as present or to be expected, behaviors that differ from when the roots are Real. I'm sure you can find many specific examples on the Internet.

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[xxMsJojoxx]

Hi Ms Jojo. We don't see imaginary zeros on xy-coordinate graphs because they are not Real numbers. Such graphs show only Real x-values and Real y-values.

You may confirm the imaginary roots by substituting them (one at a time) for symbol x in f(x) and then evaluating to obtain zero.

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Thanks! Do you know why they exist then? What is the point of having the 'imaginary roots'?

 

[Otis]

… Do you know why they exist then? …

Mathematicians are interested in finding all polynomial roots, so they want to solve for f(x)=0 even when a polynomial's graph doesn't touch or cross the x-axis. This is one reason why the imaginary unit was developed, to find all solutions (Real or not). You may google the subject, for more information.

What is the point of having the 'imaginary roots'?

That's a good research question for you! Google its variations, to find specific information like:

"Imaginary numbers are particularly applicable in electricity, specifically alternating current (AC) electronics. AC electricity changes between positive and negative in a sine wave. Combining AC currents can be very difficult because they may not match properly on the waves. Using imaginary currents and real numbers helps those working with AC electricity do the calculations and avoid electrocution."

"Imaginary numbers can also be applied to signal processing, which is useful in cellular technology and wireless technologies, as well as radar and even biology (brain waves). Essentially, if what is being measured relies on a sine or cosine wave, the imaginary number is used."

"Engineers employ imaginary roots because it makes the math easier."

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