Use the Rational Root Theorem For f a0a1xa2x2anxn Zx if an

Use the Rational Root Theorem (For f = a0+a1x+a2x^2+...+anx^n Z[x], if an 0, p/q Q is a root of f, with gcd(p,q) = 1, then q | an, p | a0) to argue that x^3 + x + 7 is irreducible in Q[x]. Use elementary calculus to argue this polynomial does have exactly one real root.

Solution

Let, p/q be a root of x^3+x+7 with gcd(p,q)=1

So from rational root theorem

p|7, q|1

Hence, p =1 or 7 and q=1

So,p/q=1 or 7

But 1 or 7 are not roots of x^3+x+7

Hence x^3+x+7 is irreducible in Q[x]

f(0)=7

f(-2)=-2^3-2+7=-3

Hence by intermediate value theorem f has a root in the interval : (-2,0)

f\'(x)=3x^2+1>0

So the slope is always positive ie the graph of the curve is strictly increasing always.

SO it is possible to cross the x axis only once. And we have proved that it does cross it once in the interval (-2,0) so that is the only real root this function has.