A quesion on establishment of nature of roots

[Akshay_Anti]

we are given an equation x⁵+x=10
. How to prove that the only root for the equation is irrational? I'm an average 12th standard student. So, please keep it low. Thanks in advance.

 

[Infinitum]

Did you try using the Rational root Theorem?

It's quite clear that there is only one real root to the equation. If you can find a contradiction that the root cannot be rational, you are done

 

[Akshay_Anti]

How to contradict? Assuming a p/q form doesn't help... the root must be of form non- terminating non-repeating decimal expansion... how to do that?

 

[Number Nine]

How to contradict? Assuming a p/q form doesn't help... the root must be of form non- terminating non-repeating decimal expansion... how to do that?

The link Infinitum posted outlines the procedure. The rational zero theorem clearly describes all possible rational roots of the equation.

 

[CellCoree]

Hello,

Thank you for your question about the nature of roots in the equation x^5 + x = 10. This is a very interesting problem and it is great to see that you are thinking about the concept of irrational roots at such a young age.

First, let's define what an irrational number is. An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction. Examples of irrational numbers include pi (π) and the square root of 2 (√2).

Now, to prove that the only root for the equation x^5 + x = 10 is irrational, we need to show that the root cannot be expressed as a fraction. To do this, we can use a proof by contradiction.

Assuming that the root is rational, we can express it as a fraction p/q, where p and q are integers and q ≠ 0. Substituting this into the equation, we get:

(p/q)^5 + p/q = 10

Multiplying both sides by q^5, we get:

p^5 + pq^4 = 10q^5

We can see that p^5 and pq^4 are both integers, so their sum must also be an integer. However, 10q^5 is not an integer since q ≠ 0. This means that our assumption that the root is rational leads to a contradiction, and therefore, the root must be irrational.

I hope this explanation was helpful and easy to understand. Keep up the curiosity and keep exploring the fascinating world of mathematics and science!

Best,