Determine which functions are rational

AI Thread Summary
Rational functions are defined as fractions of polynomials, and in this discussion, only options (i) and (iii) qualify as rational functions. Options (ii) and (iv) do not meet this criterion, as they cannot be expressed as a fraction of two polynomials. Specifically, transforming option (iv) by multiplying both the numerator and denominator does not yield a polynomial in the denominator. The discussion emphasizes the importance of the polynomial definition in identifying rational functions. Overall, the conclusion is that only (i) and (iii) are rational based on the provided definitions.
chwala
Gold Member
Messages
2,827
Reaction score
415
Homework Statement
consider the attached problem
Relevant Equations
rational functions
1628756836435.png


Ok in my thinking, i would say that it depends on ##x##, if ##x## belongs to the integer class, then the rational functions would be ##i ## and ##iii##...but from my reading of rational functions, i came up with this finding:

1628757186237.png


I would appreciate your input on this.
 
Physics news on Phys.org
Well according to the above definition only (i) and (iii) are rational functions.

(ii) or (iv) doesn't look like they are a fraction of polynomials neither I can find a way to transform them to a fraction of two polynomials. For example if we multiply both the numerator and denominator of (iv) by ##\sqrt{x-1}## we get $$\frac{x-1}{\sqrt{x^2-1}}$$ the numerator has become polynomial, but the denominator still isn't polynomial.
 
  • Like
Likes chwala and sysprog
Delta2 said:
Well according to the above definition only (i) and (iii) are rational functions.

(ii) or (iv) doesn't look like they are a fraction of polynomials neither I can find a way to transform them to a fraction of two polynomials. For example if we multiply both the numerator and denominator of (iv) by ##\sqrt{x-1}## we get $$\frac{x-1}{\sqrt{x^2-1}}$$ the numerator has become polynomial, but the denominator still isn't polynomial.
thanks, i guess i missed the term "polynomial" cheers mate:cool:
 
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

Similar threads

Back
Top