The Factor Theorem says that if f(a) = 0 then (x − a) is a factor. Roots and factors are two views of one fact.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The Rational Root Theorem
Any rational root is a factor of the constant term over a factor of the leading coefficient.
What it buys
It does not find roots. It reduces an infinite search to a finite list worth testing.
After a hit
Divide the found factor out. What remains has degree one lower and is often a quadratic you can finish.
Repeat as needed
A quartic may need two rounds before a quadratic appears. The method shrinks the problem each time.
Finishing with complex roots
When the remaining quadratic has a negative discriminant, the last two roots are a complex conjugate pair.
Roots and factors are the same information
If f(a) = 0 then (x − a) is a factor, and conversely. Every root found lets you divide it out and reduce the degree of what remains.
The rational root theorem narrows the search
Any rational root is a factor of the constant over a factor of the leading coefficient. That converts an infinite search into a short list to test, which is what makes the method practical.
Shrink the problem each time
Divide out each root found and work with the lower-degree quotient. A quintic becomes a quartic, then a cubic, until a quadratic remains that the formula finishes.
Count the roots against the degree
A degree-n polynomial has exactly n roots over the complex numbers, counted with multiplicity. Checking that your total matches the degree confirms nothing was missed.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Kofi Solve a Cubic
Kofi solves x³ − 4x² + x + 6 = 0.
- Step 1
The constant is 6 with leading coefficient 1, so candidates are ±1, ±2, ±3 and ±6.
Step 4: Your Turn
Practice makes it stick.
The Candidates
Problem 1 of 2
For x³ + 2x² − 5x − 6, how many candidate rational roots does the theorem list, counting both signs?
The Test
Problem 2 of 2
f(2) = 0. Is (x − 2) a factor? 1 yes, 0 no.
Hunt the Roots
1 of 8
x³ − x = 0. How many real roots?
2 of 8
x³ − 8 = 0. What is the real root?
3 of 8
f(3) = 0. Which factor does that give? Enter 3 for (x − 3).
4 of 8
After dividing a cubic by a linear factor, what degree remains?
5 of 8
A cubic has roots 2, 3i and one more. Imaginary coefficient of the third?
6 of 8
x³ − 6x² + 11x − 6 factors as (x−1)(x−2)(x−3). Largest root?
7 of 8
Put the solving process in order.
- 1Test candidates until one gives zero.
- 2Divide the polynomial by that factor.
- 3Solve the smaller polynomial that remains.
- 4List the candidate rational roots.
8 of 8
x³ − 4x = 0. How many real roots?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x³ − 9x = 0. How many real roots?
Question 2 of 2
What does the Rational Root Theorem actually provide?
What You Learned
- A root and a factor are the same fact, by the Factor Theorem.
- The Rational Root Theorem gives a finite list of candidates.
- Divide out each root found and solve what remains.