Question

Please read the question clearly and try to understand what actually I want to ask!

Why a polynomial can have (finite ) numbers of terms ?
I mean if numbers are infinite then why the numbers of terms in a polynomial are finite?
For example, numbers can be from 1,2,3,_ _ _ _ _ _ _ _ _. to 188907654, 188907655 or it maybe 3325789653146999........ Which is endless.
Similarly, why not terms of polynomial be like the same.I want to ask , for instances, x to the power 150 , x to the power 149...............+ x square +x+1 is a polynomial with 151 terms, then can't we say that may the terms of polynomial can began from x to the power 90987664423 ...........to+ x square +x+1 ??
Or can't we say that it may began from any number (which is infinite) ??

I suppose so many Maths Arya Bhatt is present here , so please solve my doubt with meaningful answers.
Most satisfying answer will mark as brainliest.
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Answer 1

Answer:

Hi, I'll try my best to answer your doubts here

You have asked, why can't a polynomial has infinite number of terms? Why it only has finite number of terms? Why can't the terms of the polynomials be the same?

Polynomials have only finite number of terms because that's how they are defined. Polynomials have only finite number of terms and the power (exponent) of the unknown (variable) x is only whole number. It was seen that all those algebraic expressions where the terms are finite and the power is a whole number behave in certain way, share some common properties and therefore they can be studied together. Remember one purpose of mathematics is to find some pattern amidst the randomness so that the things could be generalised.

The polynomials can have one lakh or one crore terms, they are still polynomials. Their degree is finite. If the terms are infinite, can we define the degree of the polynomials? Of course not. Yes the terms of the polynomials can begin from any finite number. They can very well start from x to the power 90987664423. In that case the degree of the polynomial will be 90987664423.

If the two terms of the polynomials are same then they can be combined to form one term. For example we have 3x^2 and 3x^2 then they both will add up to make 6x^2 and they won't be counted as two terms rather a single term.

One more thing that I would like to point out. You may not be aware of Infinite series. Do you know the expansion of log (1 +x)? Its expansion is very much like a polynomial except it has infinite number of terms.

So a polynomial with infinite number of terms is a different beast altogether. It becomes a power series and it is entirely different field of study.

Note: Expansion of log (1 + x)

$\log(1+x)=x-\frac{x^2}{2} +\frac{x^3}{3} -\frac{x^4}{4} +\frac{x^5}{5}-\frac{x^6}{6}+.........$

Hope this explanation was helpful to you.