3.2 Polynomial function  (Page 2/3)

3 : Roots having square root term occur in pairs 1+√3 and 1-√3.

4 : If a polynomial equation involves only even powers of x and all terms are positive, then all roots of polynomial equation are imaginary (complex). For example, roots of the quadratic equation given here are complex.

$$x^4+2x^2+4=0$$

Descartes rules of signs

Descartes rules are :

(i) Maximum number of positive real roots of a polynomial equation f(x) is equal to number of sign changes in f(x).

(ii) Maximum number of negative real roots of a polynomial equation f(x) is equal to number of sign changes in f(-x).

The signs of the terms of polynomial equation $f\left(x\right)=x^3+3x^2-12x+3=0$ are “+ + - +”. There are two sign changes as we move from left to right. Hence, this cubic polynomial can have at most 2 positive real roots. Further, corresponding $f\left(-x\right)=-x^3+3x^2+12x+3=0$ has signs of term given as “- + + +“. There is one sign change involved here. It means that polynomial equation can have at most one negative root.

Polynomials

Zero polynomial

The function is defined as :

$$y=\mathrm{f(x)}=0$$

The polynomial “0”, which has no term at all, is called zero polynomial. The graph of zero polynomial is x-axis itself. Clearly, domain is real number set R, whereas range is a singleton set {0}.

Constant function

It is a polynomial of degree 0. The value of constant function is constant irrespective of values of "x". The image of the constant function (y) is constant for all values of pre-images (x).

$$y=\mathrm{f(x)}=c$$

The graph of a constant function is a straight line parallel to x-axis. As “y = (f(x) = c” holds for real values of “x”, the domain of constant function is "R". On the other hand, the value of “y” is a single valued constant, hence range of constant function is singleton set {c}.We can treat constant function also as a linear function of the form f(x) = c with m=0. Its graph is a straight line like that of linear function.

There is an interesting aspect about periodicity of constant function. A polynomial function is not periodic in general. A periodic function repeats function values after regular intervals. It is defined as a fuction for which f(x+T) = f(x), where T is the period of the function. In the case of constant function, function value is constant whatever be the value of independent variable. It means that $\mathrm{f(x}+a_1)=\mathrm{f(x}+a_2)=..........\mathrm{f(x)}=c$ . Clearly, it meets the requirement with the difference that there is no definite or fixed period like "T". The relation of periodicity, however, holds for any change to x. We, therefore, summarize (it is also the accepted position) that constant function is a periodic function with no period.

Linear function

Linear function is a polynomial of order 1.

$$f\left(x\right)=a_{0}x+a_1$$

It is also expressed as :

$$f\left(x\right)=mx+c$$

The graph of a linear function is a straight line. The coefficient of “x” i.e. m is slope of the line and c is y-intercept, which is obtained for x = 0 such that f(0) = c. It is clear from the graph that its domain and range both are real number set R.