Do the values of polynomials go on infinity or stop at a certain value?

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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It depends on the functions. Some functions approaches certain value when x aporoaches inf. Example of this type is f(x)= 1/x. in this function f(x) appraches 0 as x--> inf.

However, the function f(x) = x^2 +x  approaches inf. as x-->inf.

In polynimial fractions for example f(x)= ax^2+x/bx^2+1 , f(x) approaches a/b as x--> inf.

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neela | High School Teacher | (Level 3) Valedictorian

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A polynomial is an expression of finite length formed of variables and constants  using only operations of addition, subtraction, multiplication and non-negative whole number exponents. Some examples are :

p(x) = a0+a1x+a2x^2+a^*x^n, is a polynomial is a sigle vwriable with n terms.

p(x) = x.  A polynomial with a single term . (Alsocalled monomial).

p(x) = a+bx.  Also called binomial.

The following are not examples:

p(x) = 5/x. Reason 1/x or x^(-1) has no non-negative whole number exponent.

p(x) =  x^2+ x^3/2. The second term has an exponent which is not a whole number.


"Do the value of a polynomial go on infinity ?..." Hope  you mean whether the polynomial goes on increasing and approaches infinity as x tends to infinity.

To decide whether a polynomial increases or decreases depends on the leading term (or the term with highest exponent) and its coefficient. If the coefficient of the leading term is positve, the polynomial increses otherwise it decreases along with x.

The polynomial  aproaches infinity as x --> infinity if the leading term has a positive coeffcient. It aproaches minus infinity as x-->infinity ,if the leading term has a negative coefficient. A plynomial p(x) cannot go for a definite limit when the x (or the variable) approaches plus or minus infinity.(Please do not get confused with convergence of a series for |x| <1 and  limit of the nth term a, x^n for large n). P(x) does not take a finite limit as x--> infinity (or minus infinity). unless it is polynomial with only a constant term.

Example : p(x) = x approaches infinity as x-->inf.

p(x) = x^2  - x approaches ifinity as x-->plus ifinity or x --> minus infinty, as the  term is x^2 has a positive coefficient and x ^2


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