## General Properties of Polynomials
A
A
The constants \( a_0,\dots,a_n \) in \( (\ast) \) are the
We can assume in \( (\ast) \) w.l.o.g. that \( a_n\neq0 \) (if \( a_n=0 \), the
summand \( a_nx^n \) can be erased without changing the polynomial).
Then the exponent \( n \) is called the Any two polynomials can be added, subtracted or multiplied, and the result will be a polynomial too: \begin{eqnarray*}A(x)&=&a_0+a_1x+\cdots+a_nx^n,\quad\quad\quad B(x)=b_0+b_1x+ \cdots+b_mx^m\\ A(x)\pm B(x)&=&(a_0-b_0)+(a_1-b_1)x+\cdots,\\ A(x)B(x)&=&a_0b_0+(a_0b_1+a_1b_0)x+\cdots+a_nb_mx^{m+n}.\end{eqnarray*} The behavior of the degrees of the polynomials under these operations is clear: The conventional equality \( \deg 0=-\infty \) actually arose from these properties of degrees, as else the equality (b) would not be always true. Unlike a sum, difference and product, a quotient of two polynomials is not necessarily a polynomial. Instead, like integers, they can be divided with a residue.
We say that polynomial \( A \) is
Number \( a \) is a To determine a zero of a polynomial \( f \) means to solve the equation \( f(x)=0 \). This is not always possible. For example, it is known that finding the exact values of zeros is impossible in general when \( f \) is of degree at least 5. Nevertheless, the zeros can always be computed with an arbitrary precision. Specifically, \( f(a)< 0< f(b) \) implies that \( f \) has a zero between \( a \) and \( b \).
More generally, the following simple statement holds. The converse does not hold. Although every zero of \( x^2 \) is a zero of \( x \), \( x^2 \) does not divide \( x \).
Every nonconstant polynomial with complex coefficients has a
complex root. This result is called the The following statement is analogous to the unique factorization theorem in arithmetics.
Grouping equal factors yields the
We say that two polynomials \( Q \) and \( R \) are
If polynomial \( P(x)=x^n+\cdots+a_1x+a_0 \) with real coefficients has a complex zero \( \xi \), then \( P(\overline{\xi})=\overline{\xi^n} +\cdots+a_1\overline{\xi}+a_0=\overline{P(\xi)}=0 \). Thus: In the factorization of a real polynomial \( P(x) \) into linear factors we can group conjugated complex zeros: \[ P(x)=(x-r_1)\cdots(x-r_k)(x-\xi_1)(x-\overline{\xi_1})\cdots (x-\xi_l)(x-\overline{\xi_l}),\] where \( r_i \) are the real zeros, \( \xi \) complex, and \( k+2l=n=\deg P \). Polynomial \( (x-\xi)(x-\overline{\xi})= x^2-2Re\xi+|\xi|^2=x^2-p_ix+q_i \) has real coefficients which satisfy \( p_i^2-4q_i< 0 \). This shows that: It follows that a real polynomial of an odd degree always has an odd number of zeros (and at least one). |

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