To prove by contradiction that there is no rational number $r$ such that $r^3 + r + 1 = 0$, we assume the contrary: that there exists a rational number $r = \frac{a}{b}$ in its lowest terms (where $a$ and $b$ are integers with no common factors other than 1, and $b \neq 0$) that satisfies the equation.

Starting with the equation: $r^3 + r + 1 = 0$

Substitute $r = \frac{a}{b}$: $\left(\frac{a}{b}\right)^3 + \frac{a}{b} + 1 = 0$

Multiply through by $b^3$ to clear the denominators: $a^3 + ab^2 + b^3 = 0$

Rearrange this to: $a^3 + ab^2 = -b^3$

This equation shows that $b^3$ must divide $a^3 + ab^2$. Let's analyze the parity (odd or even nature) of $a$ and $b$:

1. Case 1: Both $a$ and $b$ are even.

   If $a$ and $b$ are both even, then they share a common factor of 2. This contradicts our assumption that $\frac{a}{b}$ is in lowest terms.

2. Case 2: Both $a$ and $b$ are odd.

   If $a$ and $b$ are both odd, then $a^3$, $ab^2$, and $b^3$ are all odd (since the product of odd numbers is odd). The sum of three odd numbers is odd. Thus, $a^3 + ab^2 + b^3$ would be odd, which contradicts the equation $a^3 + ab^2 = -b^3$, since the right-hand side $-b^3$ is odd and the left-hand side should then be even.

3. Case 3: One is odd and the other is even.

   Without loss of generality, assume $a$ is odd and $b$ is even. If $b$ is even, then $b^3$ is even, and $ab^2$ is even (since it includes $b^2$). However, $a^3$ is odd. Therefore, $a^3 + ab^2$ is the sum of an odd number and an even number, which is odd. This again contradicts the fact that $-b^3$ (right-hand side) is even.

Since all possible cases lead to a contradiction, our initial assumption that $r = \frac{a}{b}$ is rational must be false. Therefore, there is no rational number $r$ such that $r^3 + r + 1 = 0$.

Would you like more details or have any questions?

Here are 5 more questions you might consider:

1. How can we use proof by induction to show a given statement?
2. What is the difference between direct proof and proof by contradiction?
3. Can you provide an example of a rational number that satisfies a polynomial equation?
4. How can we determine the roots of a polynomial equation?
5. What is the significance of the Rational Root Theorem in polynomial equations?

Tip: When tackling proof problems, always start by clearly stating your assumptions and method of proof. This helps in systematically reaching the conclusion.