What is the standard form of a polynomial function with given roots and degree?

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Homework Help Overview

The discussion revolves around finding the standard form of a polynomial function given specific roots (2, -2i) and a degree of 4. Participants explore the implications of these roots on the polynomial's structure, particularly focusing on the requirement for real coefficients.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the factors corresponding to the given roots and question the necessity of including complex conjugates. There is also debate about the interpretation of the problem statement regarding the completeness of the roots.

Discussion Status

The conversation is active, with participants providing insights about the nature of polynomial roots, particularly in relation to real coefficients and complex roots. Some guidance is offered regarding the inclusion of complex conjugates, but there is no explicit consensus on how to interpret the problem fully.

Contextual Notes

There is a noted constraint that the polynomial must have real coefficients, which influences the discussion about the roots. Participants highlight the ambiguity in the problem statement regarding whether the listed roots are exhaustive.

thomasrules
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The Roots and degree of a polynomial function are given. Write the function in standard form.

b) 2, -2i, degree 4

obviously i know there is a function with [tex]x^4[/tex] and it should have 4 x answers so I don't know how to do this...I know that (x-2) is a factor
 
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then what about (x+2i) and who says that 2 can only be a root once? What about -2i?
 
Does the polynomial have to be real? If so, note the roots must come in complex conjugate pairs. If not, just multiply together the right number of factors of (x-r), where r is a root.
 
yes it has to be real
 
then statusX is saying if -2i is a root, then 2i must be a root as well, since 2i is the complex conjugate of -2i [the complex conjugent of a+bi is a-bi, you just have no a in this situation]
 
SO IT'S

[tex](x-2)(x+2)(x+2i)(x-2i)[/tex]
 
I am confused. The problem says "The Roots and degree of a polynomial function are given" and I would interpret "The Roots" to mean that the the polynomial has only those roots. However, then it couldn't have real coefficients.

If the polynomial has real coefficients and then the roots given are not all. Clearly -2i must be another but the fourth could be any real number. There are an infinite number of solutions: (x-a)(x-2)(x2+4) where a is any real number.
 

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