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anchovy

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Oct 3, 2021
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Hi I already solved these, but can someone check my answers for me? Thanks in advance!
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The one that says choose all is not correct.

Can you please carefully define a polynomial? A polynomial of degree n will have how many roots?
 
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It will have n number of roots.
So my thinking is that it can have an infinite number of roots if n keeps going.
 
You failed to answer one of my questions! I will only ask it once more. Can you please state a precise definition of a polynomial. Also, please tell us your math background--what is the highest math you have taken?
 
A polynomial is an expression with variables and coefficients. The highest level of math I've taken is Algebra 2.
 
anchovy said:
It will have n number of roots.
So my thinking is that it can have an infinite number of roots if n keeps going.
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I think you're misinterpreting the English:

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Suppose it asked whether a country can have an infinite number of citizens. This does not mean, "Is the size of a generic country theoretically unlimited, so that a country can have as many citizens as you can imagine?"; but rather, "Is it possible for any individual country to have an infinite number of citizens?" The answer is no; any particular country has some finite number of citizens, even if there is no limit to how many there may be.

Is there any one polynomial that has infinitely many roots?
 
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Perhaps you guys are far too advanced! I ended up submitting it as is. since I was pretty confident and I got it right!
 
By definition, all polynomials have a finite degree. One can't argue with a definition.
That is, all polynomials have a highest power. Therefore, a polynomial can't have an infinite number of roots since that would mean that degree of the polynomial would be infinite.

There is the study of 'infinite' polynomials and these 'polynomials' are called power series.
 
anchovy said:
Perhaps you guys are far too advanced!
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Hi anchovy. This is not an advanced exercise, and what we are is 'accurate', heh.

All polynomials have 'degree'. Degree is a Real number. Infinity is not a Real number.

?

[imath]\;[/imath]
 
What? You got it right?!
Who said it was right? Your teacher?
I assure you that the definition of a polynomial does not allow the power of x to be infinite.
 
Yes, my teacher said I got it right! I understand the explanation that you guys are giving. But, I decided to look it up and I found this https://math.stackexchange.com/questions/1137190/is-there-a-polynomial-that-has-infinitely-many-roots/1137202#:~:text=The%20only%20polynomial%20with%20infinitely,the%20fundamental%20theorem%20of%20algebra
Doesn't make much sense to me, but I'm sure you guys know what they're talking about! I guess the polynomial f(x)=0 has infinite roots since it has no degree?
 
I would not consider f(x)=0 when talking about number of zeros of polynomials.

Suppose f(x) =0 . Then f(4)=f(.56) = f(sqrt(7))= f(2/3) = ... all equal 0. Some are saying that therefore this function f has infinitely many zeros. Again, I would exclude f(x) = 0.

Whether or not we include f(x)=0, the reason a polynomial can have an infinite number of 0 is NOT because the degree of f(x) can not be infinite.
 
Can you please ask your teacher which function has infinitely many zeros?
 
Jomo said:
Can you please ask your teacher which function has infinitely many zeros?
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That's been answered: the zero (constant) polynomial. For confirmation, see

Degree of a polynomial - Wikipedia

en.wikipedia.org en.wikipedia.org

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually undefined.​

mathworld.wolfram.com

Zero Polynomial -- from Wolfram MathWorld

The constant polynomial P(x)=0 whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials. The degree of the zero polynomial is...
mathworld.wolfram.com

The constant polynomial P(x)=0 whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials.​

That last sentence is the reason we have to consider this a polynomial!
Jomo said:
I would not consider f(x)=0 when talking about number of zeros of polynomials.

Suppose f(x) =0 . Then f(4)=f(.56) = f(sqrt(7))= f(2/3) = ... all equal 0. Some are saying that therefore this function f has infinitely many zeros. Again, I would exclude f(x) = 0.

Whether or not we include f(x)=0, the reason a polynomial can have an infinite number of 0 is NOT because the degree of f(x) can not be infinite.
Click to expand...
I feel that way, too, but I can't honestly say it.

I consider the problem to be a trick question. It's appropriate only if this has been explicitly mentioned in the class.

(In particular, when I see "infinitely many zeros", I picture a countable infinity; that isn't what this is! No wonder we were fooled.)
 
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