Show all the rational solutions of 2x3 5x2 x 10SolutionWE

Show all the rational solutions of 2x^3 + 5x^2 - x - 1=0

WE use the Rational Root Theorem

Let, x=p/q be a rational root of given equation ,gcd(p,q)=1

Then by Rational Root Theorem, p is an integer factor of -1

SO, p can be 1 or -1

q is an integer factor of 2

HEnce, q=1 or 2

So possible roots are:

1,-1,1/2,-1/2

Checking x=1

2+5-1-1=5 , so x=1 is not a root

Checking x=-1

-2+5+1-1=3 , so x=-1 is not a root

Checking x=1/2

2*(1/2)^3+5*(1/2)^2-1/2-1=1/4+5/4-3/2=0

So one root is:x=1/2

Checking x=-1/2

2*(-1/2)^3+5*(-1/2)^2+1/2-1=-1/4+5/4-1/2=1/2

So only rational solution is

x=1/2