Polynomial or Not?

College Algebra
Chapter 1/Section 4


Tell whether the expression is a polynomial. If it is, give the degree. If it's not a monomial, state the reason.

18. 1 - 4x...This is a polynomial. The degree is 1.

20. -pi...This is a polynomial. I say -pi = -pi(x^0). Since x^0 = 1, then the degree is 0.

22. (3/x) + 2...This is not a polynomial because (3/x) is actually 3x^(-1). The negative degree removes the polynomial definition.

24. 10z^2 + z...This is a polynomial. The degree is 2.

26. (3x^3 + 2x - 1)/(x^2 + x + 1)
Although the top and bottom expressions are polynomials individually, the bottom polynomial has a degree greater than 0. So, this algebraic fraction is not a polynomial.

Note: For 26, why can't a polynomial in an algebraic fraction that has a degree greater than 0 in the denominator not be a polynomial as a fraction?

 

correct

note: -pi is a monomial ( expression containing only one term) as polynomial ( of degree 0 of R[x]) as well

 

What do you mean by R[x]? Any number by itself, say 10, is actually 10x^0 = 10(1) = 10. So, pi is a monomial. It is also a polynomial of degree 0 for the reasons I stated. No???

 

R[x] is set of real numbers x
yes, it is also a polynomial of degree 0 for the reasons you stated
but if you say -pi = -pi(x^0), that is a monomial (mono=one, means one term)
but, I if you say -pi = -pi+(x^0) or -pi = -pi-(x^0) which is binomial (bi=two, so polynomial with two terms)

 

Ok but I didn't say -pi + x^0 or -pi - x^0 is a monomial. I didn't say that.

 

read carefully, I stated: you say -pi = -pi(x^0)

 

I though every number (including pi and e) is multiplied by an invisible term x^0. As you know, x^0 is 1. Take the number 4. What is the difference between 4 and 4•x^0 when x^0 is the same as 1? The answer is there is no difference. Apply that to the numbers pi and e.

 

but all these are monomials

 

Moving on....