Prove by contradiction, every real solution of x3+x+3=0 is irrational.

Prove by contradiction, every real solution of x³+x+3=0 is irrational.

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Expert Solution

Proof that every real solution of x^3+x+3=0 is irrational: (Proof by Contradiction)

Assume to the contrary there is a rational number x=p/q, in reduced form, with p and q not equal to zero, that satisfies the equation. Then, we have p³/q³ + p/q+ 3 = 0. After multiplying each side of the equation by q³, we get the equation;

p³ + p q² + 3q³ = 0

There are four cases to consider :

(1) If p and q are both odd, then the left hand side of the above equation is odd. But zero is not odd, which leaves us with a contradiction.

(2) If p is even and q is odd, then the left hand side is odd, again a contradiction.

(3) If p is odd and q is even, we get the same contradiction.

(4) If p even and q even, which is not possible because we assumed that p/q is in reduced form.

So as in four cases we got contradictory the assumption that root of 'x' is rational is false. That implies roots of 'x' in the given equation are irrational.

HENCE PROVED

Colby Messinger answered 1 year ago

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