Rational Root Theorem
Possible rational roots are ± factors of constant over factors of leading coefficient.
Key idea
Possible rational roots are ± factors of constant over factors of leading coefficient.
In depth
Any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient. This gives a finite list of candidates to test.
Worked example
2x³ − 3x² − 3x + 2: candidates ±1, ±2, ±1/2. Testing x = 2 gives 0, so it is a root.
Formula
Real-world uses
- Finding exact roots of polynomial models in chemistry and economics with a small list of candidates
Common mistake
Forgetting the ± signs or skipping candidates.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rational Root Theorem?
Possible rational roots are ± factors of constant over factors of leading coefficient. It is a topic at the expert level in Algebra.
What is the formula for Rational Root Theorem?
The key formula is p/q.
Where is Rational Root Theorem used in real life?
Real-world examples: Finding exact roots of polynomial models in chemistry and economics with a small list of candidates.
What should I learn before Rational Root Theorem?
Make sure you are comfortable with Remainder and Factor Theorems, the previous topic in Algebra.
What is a common mistake with Rational Root Theorem?
Forgetting the ± signs or skipping candidates.
How can I practise Rational Root Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rational Root Theorem?
Move on to Fundamental Theorem of Algebra next.
Related expert topics in Algebra
- Linear Programming – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
- Fundamental Theorem of Algebra – Every degree-n polynomial has exactly n complex roots counted with multiplicity.
- Binomial Theorem – Expand (a+b)ⁿ with binomial coefficients.
- Partial Fractions – Decompose a rational function into simpler fractions.
- Mathematical Induction – Prove statements for all n by base case and inductive step.
- Cramer’s Rule – Solve linear systems using determinants.