description/proof of that real polynomial is divided into 1-degree real polynomials and 2-degree real polynomials
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of polynomials ring over commutative ring.
- The reader admits the fundamental theorem of arithmetic.
- The reader admits the proposition that for any field and any subfield, any polynomial and any nonzero polynomial divisor over the field both with only subfield coefficients have the quotient and the remainder both with only subfield coefficients.
Target Context
- The reader will have a description and a proof of the proposition that any real polynomial is divided into some 1-degree real polynomials and some 2-degree real polynomials.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R} [x]\): \(= \text{ the polynomials ring over } \mathbb{R}\)
\(p (x)\): \(\in \mathbb{R} [x]\), \(= r_n x^n + ... + r_0\) where \(r_n \neq 0\)
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Statements:
\(p (x)\) is divided into some 1-degree real polynomials and some 2-degree real polynomials
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2: Proof
Whole Strategy: iteratively factor out a 1-degree or 2-degree real polynomial; Step 1: when \(p (x)\) is equal-to-or-smaller-than-\(2\)-degree, see that we do not need to do anything for it; Step 2: when \(p (x)\) is larger-than-\(2\)-degree, regard \(p (x)\) as a complex polynomial, and take \(p (x) = r_n (x - c_1) ... (x - c_n)\), by the fundamental theorem of arithmetic; Step 3: see that \(c_1\) is real or there is a \(c_j = \overline{c_1}\), and see that \(p (x)\) divided by \(r_n (x - c_1)\) or \(r_n (x - c_1) (x - c_j)\) is a real polynomial, \(q (x)\); Step 4: iteratively do likewise for \(q (x)\).
Step 1:
When \(p (x)\) is equal-to-or-smaller-than-\(2\)-degree, we do not need to do anything for it.
Step 2:
Let us suppose that \(p (x)\) is larger-than-\(2\)-degree.
\(p (x)\) can be regarded to be a complex polynomial.
By the fundamental theorem of arithmetic, \(p (x) = r_n (x - c_1) ... (x - c_n)\) where \(c_1, ..., c_n\) are possibly duplicate complex numbers.
Step 3:
If \(c_1\) is real, \(q (x) := (x - c_2) ... (x - c_n)\) is a real polynomial, because it is the quotient of \(p (x)\) divided by \(r_n (x - c_1)\), by the proposition that for any field and any subfield, any polynomial and any nonzero polynomial divisor over the field both with only subfield coefficients have the quotient and the remainder both with only subfield coefficients.
If \(c_1\) is not real, there is a \(c_j = \overline{c_1}\) where \(\overline{c_1}\) is the complex conjugate, because as \(p (c_1) = 0\), \(p (\overline{c_1}) = \overline{p (c_1)} = 0\), because any multiplication of complex conjugates is the conjugate of the multiplication and any addition of complex conjugates is the conjugate of the addition.
\((x - c_1) (x - c_j)\) is a real 2-degree polynomial, because \((x - c_1) (x - c_j) = (x - c_1) (x - \overline{c_1}) = x^2 - x (c_1 + \overline{c_1}) + c_1 \overline{c_1}\), where \(c_1 + \overline{c_1}\) is real and \(c_1 \overline{c_1} = \vert c_1 \vert^2\) is real.
\(q (x) := (x - c_2) ... \widehat{(x - c_j)} ... (x - c_n)\) (\(\widehat{(x - c_j)}\) means that \((x - c_j)\) is missing) is a real polynomial, because it is the quotient of \(p (x)\) divided by \(r_n (x - c_1) (x - c_j)\), by the proposition that for any field and any subfield, any polynomial and any nonzero polynomial divisor over the field both with only subfield coefficients have the quotient and the remainder both with only subfield coefficients.
Step 4:
If \(q (x)\) is equal-to-or-smaller-than-\(2\)-degree, we do not need to do anything for it.
Otherwise, we do for \(q (x)\) likewise instead of for \(p (x)\).
And so on, after all, \(q (x)\) becomes equal-to-or-smaller-than-\(2\)-degree.
Thus, \(p (x)\) is divided into some 1-degree real polynomials and some 2-degree real polynomials.
