Publish On: 2019-03-18

# Ben joy

Total Post: 542

## Question: What is rational and integral polynomial

Reply On: 2013-10-07

# Mjay Jollay

Total Post: 0

## ANS: What is rational and integral polynomial

An expression of the form ƒ(x) = a

(i) They are of the same degree, i.e. m = n.

(ii) Their corresponding coefficients are equal, i.e. a

For example, consider the following equations:

x

The equation (i) when expressed in rational and integral form is 6x

9x

_{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).**Degree of polynomial:**If a_{0 }≠ 0, then the polynomial 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}xn + a_{1}x^{n-1 }+ …. + a_{n}will be an identically vanishing polynomial if ai = 0, ∀ 1 ≤ i ≤ n.**Note 1:**No degree is assigned to an identically vanishing polynomial.**Note 2:**Constants (other than zero) are polynomial of degree zero.**Equality of polynomials:**Two polynomials ƒ(x) = a_{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, are said to be equal if(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_{n}, a_{0}≠ 0, will be an equation if ƒ(x) = a_{0}x^{n}+ a_{1}x^{n-1}+ …. + a_{n}, a_{0}= 0, for some values of 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:

x

^{-3/2}+ 4x^{2}= 3x^{1/2}The equation (i) when expressed in rational and integral form is 6x

^{8}+ 5x^{6}+ 6x^{5}+ 6x^{2}+ 1 = 0 and thus its degree is 8 and the equation (ii) when expressed in rational integral form is of degree 3 for9x

^{3}– 6x^{2}+ x – 16 = 0.**Note 3:**The equation of degree two, three and four are called quadratic, cubic and biquadratic respectively.Like Us On Facebook for All Latest Updates