Question 1171009: I don't need the solution to the following question. I just need someone to explain to me what is the question!...more of a language problem. If after understanding the question and if I am unable to solve the problem I will repost the question.
Q: Write the polynomial equation of lowest degree with 'constant (real or complex) coefficients' having the roots 4 and 2-3i.
What is a constant coefficient? Constant in the polynomial is the coefficient of x^0. With the given, the factors are (x-4),(x-2+3i),(x-2-3i)
Thank you
Answer by ikleyn(52817)   (Show Source):
You can put this solution on YOUR website! .
x^2 + 3x - 4 is a polynomial of the degree 2 with constant coefficients
1 (at x^2), 3 (at x) and -4 (the constant term).
x^3 - 6x^2 + 7x - 1 is a polynomial of the degree 3 with constant coefficients
1 (at x^3), -6 (at x^2), 7 (at x) and -1 (the constant term).
-2x4 - 14x^3 + 5x^2 + 19x + 3 is a polynomial of the degree 4 with constant coefficients
-2 (at x^4), -14 (at x^3), 5 (at x^2), 19 (at x) and 3 (the constant term).
After looking into these examples, it should be clear to you what the term "coefficients of a polynomial"
or the "constant coefficients of a polynomial" means.
The coefficients in the examples above are integer numbers.
But they can be any real numbers or even complex numbers.
I hope that after my explanations the subject is a bit more clear to you.
If your interests on the subject go/stretch farther, you may read this Wikipedia article
https://en.wikipedia.org/wiki/Polynomial
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Comment from student: Ref: Question 1171009 Thank you for your help. I did look up the reference you provided as well.
In the context of polynomials, I am familiar with what the coefficient means and what Integral, Real, Rational,
and complex coefficients mean. I am still unclear about the "Constant coefficient". Is it just another
or complete name for the Coefficients of the variables that can be Integral, Real, Rational or complex?
If that is so, then the answer (please ref the question that I posed) in the book goes as follows:
(x-2)[x-(1-3i)]=0 So what happened to [x-(1+3i)] ? I suspect it has to do with the word 'Constant'? Please help.
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My response : Thank you for asking.
First, I am very satisfied that you do refer to the Problem's ID number.
It tells me that you are an responsible person.
Next, couple of words about the terminology.
When we consider polynomials, there are two major entities: there is variable " x " with its degrees; it is considered
(in accordance with its name) as variable value, which may have different values;
and there are "coefficients" that are considered as constant values for each individual polynomial.
Next issue is about the polynomial from the Problem's question and about its factors.
The question is : "Q: Write the polynomial equation of lowest degree with 'constant (real or complex) coefficients'
having the roots 4 and 2-3i."
There are two VERY DIFFERENT, I'd say, "opposite" cases here:
a) when the requested polynomial is considered over the set of complex numbers ("with complex coefficients"), and
b) when the polynomial is considered over the set of real numbers ("with real coefficients").
The polynomial of the LOWEST degree with COMPLEX coefficients (case a) having the given roots 4 and 2-3i
is the polynomial (x-4)*(x-(2-3i)). It is the product of two binomials associated with the roots.
It is the quadratic polynomial, and if you want to write it in the standard form, you should multiply
these two linear binomial with complex coefficients (making FOIL).
The polynomial of the LOWEST degree with REAL coefficients (case b) having the roots 4 and 2-3i, NECESSARY
has the complex conjugate root to the number 2-3i, which is 2+3i.
Therefore, in this case, such a polynomial has, actually, three roots 4, 2-3i and 2+3i.
And as such, it is the polynomial of the 3rd degree (x-4)*(x-(2-3i))*(x-(2+3i)).
It is the product of THREE binomials associated with the THREE roots.
Again, to write it in the standard form, you should multiply (FOIL) these 3 binomials.
These are the facts from Algebra.
These facts are proved in courses of Abstract Algebra of the University level.
For high school students these facts only announced (without formal proofs).
So, it seems to me that I answered all your questions.
As you understand, it is not my goal to bring/(to provide) a full course of Abstract Algebra here,
so I should to limit myself by these short explanations.
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