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Rational Root Theorem

Lesson · Intermediate

Algebra

Suppose

P(x)=anxn+an−1xn−1+⋯+a1x+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

has integer coefficients, and suppose

r=pqr=\frac{p}{q}

is a rational root written in lowest terms:

gcd⁡(p,q)=1\gcd(p,q)=1

Then the Rational Root Theorem gives

p∣a0p\mid a_0

and

q∣anq\mid a_n

Therefore, every rational root must have the form

±factor of a0factor of an\pm\frac{\text{factor of }a_0}{\text{factor of }a_n}

But in contest problems, we can often use the equation itself to get additional divisibility conditions and greatly reduce the number of candidates.

Key Idea

1. Use the Rational Root Theorem
2. Look for additional divisibility information from the equation
3. Reduce the list of possible roots
4. Test the remaining candidates

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