Video Transcript
Solve the cubic equation 𝑥 cubed plus 10 equals two in ℚ.
We are being asked to find a rational value of 𝑥 that makes this equation true, if such a number exists. The first thing to do is isolate the 𝑥 cubed term on the left-hand side of the equality symbol. We can do this by subtracting 10 from both sides of the equation. Next, we want to extract cube roots on both sides.
What we want is a rational number which when multiplied by itself three times yields negative eight. And that’s just negative two, right? Certainly, negative two does cube to negative eight. One way to see that no other rational number cubes to negative eight is by using the fact that the rational numbers are ordered. Every rational number that is not equal to negative two is either greater than it or less than it.
We should be able to convince ourselves that if a number is less than negative two, then its cube must be less than and therefore not equal to negative eight. Similarly, if a number is greater than negative two, then the cube will be greater than negative eight. Of course, this leaves open the possibility that there exist some other types of numbers, which are not rational and not ordered, which also cube to negative eight. And indeed, there are two such numbers. But they’re secret, so I can’t tell you about them.
