A polynomial of degree n has exactly n roots, counting repeats and complex ones. A cubic always has three.
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 divided by a factor of the leading coefficient.
What it buys you
It does not find roots. It shortens an infinite search to a finite list worth testing.
After a hit
Once a root works, divide it out. What remains has degree one lower and is often a quadratic you can finish.
Repeated roots
A root can appear more than once. At an even multiplicity the graph touches the axis and turns back; at an odd one it crosses.
Complex roots
With real coefficients, complex roots come in conjugate pairs, so an odd-degree polynomial always has at least one real root.
The rational root theorem narrows the search
Any rational root of a polynomial with integer coefficients is a factor of the constant term over a factor of the leading coefficient. That turns an infinite search into a short finite list to test.
Each root found reduces the degree
Once you find a root, divide it out and work with the lower-degree quotient. A quintic becomes a quartic, then a cubic, until you reach a quadratic you can finish with the formula.
The fundamental theorem of algebra
Every polynomial of degree n has exactly n roots, counted with multiplicity, over the complex numbers. It guarantees the roots exist; it does not tell you how to find them, and for degree five and above no general formula exists.
Report every root
A cubic has three roots. If you find one real root and stop, the answer is incomplete — the remaining quadratic may give two more real roots or a complex pair. Counting the roots against the degree is the check.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Solve x³ − 4x² + x + 6 = 0
Priya needs all three roots of this cubic.
- Step 1
The constant is 6 and the leading coefficient is 1, so the 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 = 0, how many candidate rational roots does the theorem list, counting both signs?
The Box
Problem 2 of 2
A box has volume x³ − 6x² + 11x − 6 cubic units, which factors as (x−1)(x−2)(x−3). What is the largest root?
Find Every Root
1 of 8
How many roots does a degree-5 polynomial have, counting repeats and complex ones?
2 of 8
x³ − x = 0. How many real roots?
3 of 8
(x − 4)²(x + 1) = 0. What is the multiplicity of the root 4?
4 of 8
A cubic has roots 2, 3i and one more. What is the third root, as a coefficient of i?
5 of 8
x³ − 8 = 0. What is the real root?
6 of 8
A quartic has three real roots, one of them repeated once more. How many complex roots does it have?
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
Which are candidate rational roots of x³ + 2x² − 5x − 6?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x³ − 4x = 0. How many real roots?
Question 2 of 2
Why does every odd-degree polynomial have at least one real root?
What You Learned
- A degree-n polynomial has exactly n roots, counting repeats and complex ones.
- The Rational Root Theorem turns an infinite search into a short list.
- Divide out each root you find and solve what remains.