Determine whether each expression is a polynomial.

Question

abbles · Answer

$$f(x) = 3x + 4/3x^3 - 7$$
$$g(x) = -5x^3 + x - 3/x$$
$$h(x) = x^3\sqrt{5} + x - 3$$
$$k(x) = 25x^2 - x^3 + x^5$$

abbles · Answer

Is k(x) the only polynomial?

abbles · Answer

I know g(x) is not because it's dividing by an exponent.  I know k(x) is a polynomial.  A bit confused on the other two.

phi · Answer

h(x) is also legit.  You can have "irrational coefficients" such as sqr(5)

abbles · Answer

Okay thanks!  So h(x) and k(x) are the only polynomials?

phi · Answer

yes.  the first one is a fraction of two polynomials and does not simplify to just a polynomial

abbles · Answer

The first one actually looks like this: $$f(x) = 3x + (4/3)x^3 - 7$$

abbles · Answer

I was under the impression that you couldn't have division in polynomials, but I'm not 100% sure

phi · Answer

you can, but we don't get a polynomial as an answer.  But it's easy to divide things. You write / in between them:  4/2  for example.

abbles · Answer

So why is f(x) not a polynomial exactly?

abbles · Answer

$$f(x) = f(x)=3x+(4/3)x^3−7$$

The 4/3 could be divided to 1.33 right?

phi · Answer

really 1.333.... (the 3's go on forever)
but you can leave it 4/3 which is a number.
number * x^integer  is what you need, and that is what you have

abbles · Answer

So f(x) IS a polynomial?

Sorry, I'm a little lost :P

phi · Answer

is that what the first choice is ?
I thought it was
$$ \frac{3x+4}{3x^3−7}$$

abbles · Answer

No no, I typed it wrong the first time.  See my comment above

phi · Answer

in that case it is a polynomial, though it is not in "standard form" 
(the terms go from highest exponent to lowest)

It is a bit peculiar to write it that way. I would complain to the teacher.

abbles · Answer

We are also practicing converting to standard form, so maybe that has something to do with it.

abbles · Answer

So g(x) is the only one that is not a polynomial, correct?

phi · Answer

yes

abbles · Answer

thank you