General Properties of Polynomials

A monomial in variable \( x \) is an expression of the form \( cx^k \), where \( c \) is a constant and \( k \) a nonnegative integer. Constant \( c \) can be e.g. an integer, rational, real or complex number.

A polynomial in \( x \) is a sum of finitely many monomials in \( x \). In other words, it is an expression of the form \[ P(x)=a_nx^n+a_{n-1}+\cdots+a_1x+a_0.\quad\quad\quad\quad\quad(\ast)\] If only two or three of the above summands are nonzero, \( P \) is said to be a binomial and trinomial, respectively.

The constants \( a_0,\dots,a_n \) in \( (\ast) \) are the coefficients of polynomial \( P \). The set of polynomials with the coefficients in set \( A \) is denoted by \( A[x] \) - for instance, \( \mathbb{R}[x] \) is the set of polynomials with real coefficients.

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 degree of polynomial \( P \) and denoted by \( \deg P \). In particular, polynomials of degree one, two and three are called linear, quadratic and cubic. A nonzero constant polynomial has degree 0, while the zero-polynomial \( P(x)\equiv0 \) is assigned the degree \( -\infty \) for reasons soon to become clear.

Example
 

\( P(x)=x^3(x+1)+(1-x^2)^2=2x^4+x^3-2x^2+1 \) is a polynomial with integer coefficients of degree 4.

\( Q(x)=0x^2-\sqrt2x+3 \) is a linear polynomial with real coefficients.

\( R(x)=\sqrt{x^2}=|x| \), \( S(x)=\frac 1x \) and \( T(x)=\sqrt{2x+1} \) are not polynomials.

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:

Theorem 1.1
 

If \( A \) and \( B \) are two polynomials then:

  • (a) \( \deg(A\pm B)\leq\max(\deg A,\deg B) \), with the equality if \( \deg A\neq\deg B \).

  • (b) \( \deg(A\cdot B)=\deg A+\deg B \).

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.

Theorem 1.2
 

Given polynomials \( A \) and \( B\neq0 \), there are unique polynomials \( Q \) (quotient) and \( R \) (residue) such that \[ A=BQ+R\quad\mbox{and}\quad\deg R< \deg B.\]

Example
 
The quotient upon division of \( A(x)=x^3+x^2-1 \) by \( B(x)=x^2-x-3 \) is \( x+2 \) with the residue \( 5x+5 \), as \[ \frac{x^3+x^2-1}{x^2-x-3}=x+2+\frac{5x+5}{x^2-x-3}\:.\]

We say that polynomial \( A \) is divisible by polynomial \( B \) if the remainder \( R \) when \( A \) is divided by \( B \) equal to 0, i.e. if there is a polynomial \( Q \) such that \( A=BQ \).

Theorem 1.3 (Bezout’s theorem)
 

Polynomial \( P(x) \) is divisible by binomial \( x-a \) if and only if \( P(a)=0 \).

Number \( a \) is a zero (or root) of a given polynomial \( P(x) \) if \( P(a)=0 \), i.e. \( (x-a)\mid P(x) \).

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 \( b \).

Example
 

Polynomial \( x^2-2x-1 \) has two real roots: \( x_{1,2}=1\pm\sqrt2 \).

Polynomial \( x^2-2x+2 \) has no real roots, but it has two complex roots: \( x_{1,2}=1\pm i \).

Polynomial \( x^5-5x+1 \) has a zero in the interval \( [1.44,1.441] \) which cannot be exactly computed.

More generally, the following simple statement holds.

Theorem 1.4
 

If a polynomial \( P \) is divisible by a polynomial \( Q \), then every zero of \( Q \) is also a zero of \( P \).

The converse does not hold. Although every zero of \( x^2 \) is a zero of \( x \), \( x^2 \) does not divide \( x \).

Problem 1
 
For which \( n \) is the polynomial \( x^n+x-1 \) divisible by (a) \( x^2-x+1 \), (b) \( x^3-x+1 \)?

Every nonconstant polynomial with complex coefficients has a complex root. This result is called the fundamental theorem of algebra and we will prove it later. For now, we are going to take it for granted and explore some of its consequences.

The following statement is analogous to the unique factorization theorem in arithmetics.

Theorem 1.5
 

Polynomial \( P(x) \) of degree \( n> 0 \) has a unique representation of the form \[ P(x)=c(x-x_1)(x-x_2)\cdots(x-x_n),\] not counting the ordering, where \( c\neq0 \) and \( x_1,\dots,x_n \) are complex numbers, not necessarily distinct.

Therefore, \( P(x) \) has at most \( \deg P=n \) different zeros.

Corollary
 

If polynomials \( P \) and \( Q \) has degrees not exceeding \( n \) and coincide at \( n+1 \) different points, then they are equal.

Grouping equal factors yields the canonical representation: \[ P(x)=c(x-a_1)^{\alpha_1}(x-a_2)^{\alpha_2}\cdots (x-a_k)^{\alpha_k},\] where \( \alpha_i \) are natural numbers with \( \alpha_1+\cdots+\alpha_k=n \). The exponent \( \alpha_i \) is called the multiplicity of the root \( a_i \). It is worth emphasizing that:

Theorem 1.6
 

Polynomial of \( n \)-th degree has exactly \( n \) complex roots counted with their multiplicities.

We say that two polynomials \( Q \) and \( R \) are coprime if they have no roots in common; Equivalently, there is no nonconstant polynomial dividing them both, in analogy with coprimeness of integers. The following statement is a direct consequence of the previous theorem:

Theorem 1.7
 
If a polynomial \( P \) is divisible by two coprime polynomials \( Q \) and \( R \), then it is divisible by \( Q\cdot R \).

Remark: This can be shown without using the existence of roots. By the Euclidean algorithm applied on polynomials there exist polynomials \( K \) and \( L \) such that \( KQ+LR=1 \). Now if \( P=QS=RT \) for some polynomials \( R,S \), then \( R(KT-LS)=KQS-LRS=S \), and therefore \( R\mid S \) and \( QR\mid QS=P \).

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:

Theorem 1.8
 

If \( \xi \) is a zero of a real polynomial \( P(x) \), then so is \( \overline{\xi} \).

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:

Theorem 1.9
 
A real polynomial \( P(x) \) has a unique factorization (up to the order) of the form \[ P(x)=(x-r_1)\cdots(x-r_k) (x^2-p_1x+q_1)\cdots(x^2-p_lx+q_l),\] where \( r_i \) and \( p_j,q_j \) are real numbers with \( p_i^2< 4q_i \) and \( k+2l=n \).

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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