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Rational, Integral Polynomial

An expression of the form

For example, consider the following equations:

The equation

**ƒ(x) = a**_{0}x^{n}+ a_{1}x^{n-1}+ …. + a_{n-1}+ a_{n}, where**(i) a**_{0}, a_{1}, …., a_{n}are constants, real or imaginary,**(ii) x**is a variable, and**(iii) n**is a positive integer, is called a rational integral function of**x**or a polynomial in**x**. If the coefficients**a**_{0}, a_{1}, a_{2}, …., a_{n}are real number then the equation is called an equation with real coefficients.**Coefficients:**The constants**a**_{0}, a_{1}, ….., a_{n}are called the coefficients of the polynomial of**ƒ(x)**.**If**

Degree of polynomial:Degree of polynomial:

**a**, then the polynomial_{0}≠ 0**ƒ(x)**is of degree**n**, i.e. it is the highest power of the variable**x**in the polynomial.**Identically vanishing polynomial:**A polynomial all of whose coefficients are equal to zero is called an identically vanishing polynomial and is represented by**0**, i.e. the polynomial**a**_{0}x^{n}+ a_{1}x^{n-1}+ …. + a_{n}will be an identically vanishing polynomial if**a**._{i}= 0, ∀ 1 ≤ i ≤ n**No degree is assigned to an identically vanishing polynomial.**

Note 1:Note 1:

**Note 2:**Constants (other than zero) are polynomial of degree zero.**Equality of polynomials:**Two polynomials**ƒ(x) = a**, are said to be equal if_{0}x^{n}+ a_{1}x^{n-1}+ …. + a_{n}, a_{0}≠ 0 and g(x) = b_{0}x^{m}+ b_{1}x^{m-1}+ …. + b_{m}b_{0}≠ 0**(i)**They are of the same degree, i.e.**m = n**.**(ii)**Their corresponding coefficients are equal, i.e.**a**._{i}= b_{i}, ∀ 1 ≤ i ≤ m**Equation:**If two different polynomials in the same variable**x**become equal for some values of**x**or a polynomial is equated to zero, then such a relation is called an equation, i.e. the polynomial**ƒ(x) = a**_{0}x^{n}+ a_{1}x^{n-1}+ …. +**a**, will be an equation if_{n}, a_{0}≠ 0**ƒ(x) = a**, for some values of_{0}x^{n}+ a_{1}x^{n-1}+ …. + a_{n}, a_{0}= 0**x**.**Degree of an equation:**The degree of an equation is the highest power of the variable in the equation, when it is expressed as a rational and integral function of the variable.For example, consider the following equations:

**(ii) x**^{-3/2}+ 4x^{2}= 3x^{1/2}The equation

**(i)**when expressed in rational and integral form is**6x**and thus its degree is^{8}+ 5x^{6}+ 6x^{5}+ 6x^{2}+ 1 = 0**8**and the equation**(ii)**when expressed in rational integral form is of degree**3**for**9x**^{3}– 6x^{2}+ x – 16 = 0.**The equation of degree two, three and four are called quadratic, cubic and biquadratic respectively.**

Note 3:Note 3:

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