What is difference between polynomial and rational expresions?

Question

I wondering about this term that what is difference betweem these....
and also explain this line...

"Every polynomial is regareded as a rational expressions,but every rational expression need not be a polynomial"

Prove this with examples.

Answer 1

POLYNOMIAL - The EXPONENT of the variable is a WHOLE Number {0,1,2,3,....}



P(x) = 6x^3 - 2x + 3



No variables in the Denominator. No variables under a Radical.



RATIONAL FUNCTION - The EXPONENT of the variable is NEGATIVE or a FRACTION



R(x) = 3x^(-2) + 5x - 3



R(x) = 3 / x^2 + 5x - 3



R(x) = 5√x + 6

Answer 2

A polynomial is a linear algebraic expression of a certain variable, say of 'x' , e.g.,



f(x) = a x^n + b x^(n-1) + c x^(n-2) +........................+ k x^2 + l x + m



where a, b, c, .......l, m, n are constants positive or negative or may be some of them zero, is called a polynomial of degree 'n; (a finite number) in 'x' provided 'a' is not equal to zero. In general, linear, quadratic, cubic, quartic, .... expressions are all Polynomials with specific names depending on the degree of the polynomial. Even a constant may be called a Polynomial of degree '0'.



But a rational expression is the quotient of two Polynomials f(x) and g(x) related by



[f(x) / g(x)]. For example,



( x^3 + 3)

[f(x) / g(x)]. = [ ---------------- ]

(x^2 + 1 )



when g(x) becomes a constant what remains is a polynomial. For example,



(a x^2 + b x + c)/73 is a rational expression turned into a polynomial of degree '2'. So, every polynomial can be treated as a rational fraction when g(x) = 1, or a constant.



But



73/ (a x^2 + b x + c) is strictly a Rational fraction which may be expressed in an infinite series but not as a Polynomial as defined earlier

.

Thus comes your quote :"Every polynomial is regarded as a rational expressions,but every rational expression need not be a polynomial"

Answer 3

Polynomial: any number that forms into ax^n where x is Real number

Example: 5,6a

Rational Expressions: Polynomial/Polynomial

Example: a/b where b not=0

Note: there must be no radical sign in a variable

Answer 4

Considering only one variable,x, a polynomial is of the form

f(x)=a0+a1x+a2x^2+a3x^3+....+anx^n where a0, a1,a2,...,an are constants.

A rational expression in x would be f(x)/g(x) where f and g are polynomials.

This includes the case where g(x)=c (where c is a constant) giving f(x)/c which is a polynomial.

When g(x) is not a constant then f(x)/g(x) is not a polynomial.

Example: f(x)=2+3x-5x^2 is a polynomial, h(x)= (2+3x-5x^2)/(3+2x) is a rational expression.

j(x)=(2+3x-5x^2)/3 is rational and = (2/3)+x-(5/3)x^2 which is a polynomial.

Answer 5

Polynomial is one that has more than one elements.

a. x+y

b. x+x^2

c. x-yx+xz+x^2

......... etc etc.



Rational is one in the form of a/b, p/q etc.

The enumerator and the denominator can be single element or polynomial.



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