Connor says, 'Golly! My favourite real, non-zero polynomial r(x)r(x) is divisible by q(x)q(x)

Connor says, 'Golly! My favourite real, non-zero polynomial r(x)r(x) is divisible by q(x)q(x), and in fact it has the smallest degree possible for a polynomial with those properties.'What is the degree of r(x)r(x)?

Answer in video and below: Solving the Mysterious Polynomial Puzzle: What's the Degree of r(x)r(x)? #mathchallenge #polynomials

https://youtube.com/shorts/r1HdujUHAjU?feature=share

Solving the Polynomial Degree Problem

Let's analyze what we know:

1. We have a polynomial r(x)r(x) that is divisible by q(x)q(x)

2. r(x)r(x) has the smallest possible degree for a polynomial with this property

First, I need to understand what r(x)r(x) means. This notation suggests that we're dealing with r(x) multiplied by itself. 

For r(x)r(x) to be divisible by q(x)q(x), we need r(x)r(x) to be a multiple of q(x)q(x).

The simplest way to achieve this is if r(x) = q(x), because then:

r(x)r(x) = q(x)q(x)

This would make r(x)r(x) exactly divisible by q(x)q(x), and it would have the smallest possible degree for a polynomial with this property.

Therefore, the degree of r(x)r(x) would be twice the degree of q(x).

However, since we don't know the degree of q(x), we can say:

If deg(q(x)) = n, then deg(r(x)r(x)) = 2n

The answer is 2n, where n is the degree of q(x).

Actually, since we're told r(x)r(x) has the "smallest degree possible" for a polynomial with these properties, and r(x) = q(x) gives us r(x)r(x) = q(x)q(x), the degree of r(x)r(x) is exactly equal to the degree of q(x)q(x), which is 2n.

Therefore, the degree of r(x)r(x) is 2n.