# Proving that solutions of a equation are irrational

• MHB
• ranga519
In summary, to prove that the solutions of the equation x^3 + x + 1 = 0 are irrational, we can use the Rational Root Theorem and prove by contradiction. This is similar to the proof of the irrationality of √2. Assuming that x is rational and using the equation, we can show that it leads to a contradiction, proving that there are no rational roots.
ranga519
how to prove that solutions of the following equation are irrational

x^3 + x + 1 = 0

ranga519 said:
how to prove that solutions of the following equation are irrational

x^3 + x + 1 = 0

Hi Ranga,

I do not know what your course material says, but we have the Rational Root Theorem to work with.
Since neither $x=-1$ nor $x=1$ is a solution, that implies that there is no rational root.

You can prove it by contradiction. It is essentially the same proof as the proof of $\sqrt 2$ being irrational.

Assume x is rational, then x can be given as $x=p/q$ where p and q have no common primefactors. The equation becomes $$\frac{p^3}{q^3}+\frac{p}{q}+1=0 \Leftrightarrow p^3=-q^2(p+q)$$
Can you se the contradiction now?

## 1. What is an irrational number?

An irrational number is a real number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a fraction with a finite number of digits after the decimal point.

## 2. How can you prove that a solution of an equation is irrational?

One way to prove that a solution of an equation is irrational is by contradiction. This involves assuming that the solution is rational and then showing that this leads to a contradiction. This contradiction then proves that the solution must be irrational.

## 3. Can all solutions of an equation be irrational?

No, not all solutions of an equation are irrational. Some equations may have rational solutions, such as 2x + 4 = 6, which has a rational solution of x = 1.

## 4. Are irrational numbers useful in real life?

Yes, irrational numbers are very useful in real life. They are used in various fields such as mathematics, physics, and engineering. For example, pi (π) is an irrational number that is used in many calculations involving circles and curves.

## 5. How can knowing that a solution is irrational help us?

Knowing that a solution is irrational can help us in various ways. It can help us simplify equations, make more accurate calculations, and understand the relationship between rational and irrational numbers. It can also help us solve problems that involve irrational numbers, such as finding the side length of a square with a given area.

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