Can you prove this?

Lrrgolas

x^5+x=10

How can you prove that the positive root of this equation is irrational?

(hints appreciated :smile:

)

Reply 1

Ano123

Factorise the LHS completely, leave the RHS side as it is.
Contradiction.

No this won't work.

(edited 7 years ago)

Reply 2

Math12345

Show the root is positive by using the change of sign test.

Now assume the root is rational i.e. x=a/b where a and b are integers and are coprime.

So a^5/b^5+a/b=10
a^5+ab^4=10b^5

a(a^4+b^4)=10b^5

a^5=10b^5-ab^4
a^5=b(10b^4-ab^3)

So a must divide 10 since we assumed a and b are coprime and similarly b must divide 1 (we use the fundamental theorem of arithmetic here).

So x=±1,±2,±5,±10.
You can check none of these are roots of the equation (In fact if you use the change of sign test then you only need check 2 of these). So the root must be irrational.

(edited 7 years ago)

Reply 3

1420787

General solution for a quadratic equation. Look at the part in the square root. The square root of a prime number is always irrational, and an irrational number multiplied by a rational number is always irrational