Test possible rational roots
Provide integer coefficients from the highest power down to the constant term.
Rational Zero Theorem Formula
p is any factor of a0. q is any factor of an. Test both positive and negative reduced fractions.
The calculator evaluates each candidate with Horner’s method. A zero remainder confirms the root. Synthetic division then produces the next quotient.
Calculate Possible Rational Roots
- Select the highest exponent of your polynomial.
- Enter coefficients from the highest power to the constant.
- Keep zero coefficients for missing terms.
- Choose a tolerance, display precision, and graph range.
- Press “Find rational zeros” to view candidates and factors.
- Download CSV for spreadsheet work or PDF for sharing.
Worked Polynomial Example
| Polynomial | Constant factors p | Leading factors q | Possible candidates | Verified zeros |
|---|---|---|---|---|
| x3 − 6x2 + 11x − 6 | 1, 2, 3, 6 | 1 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| 2x2 − 3x + 1 | 1 | 1, 2 | ±1, ±1/2 | 1, 1/2 |
| x3 − 4x | 0 | 1 | 0, then quotient candidates | 0, −2, 2 |
Understand Rational Candidates
The theorem helps inspect polynomial roots before factoring. It needs integer coefficients. It lists possible rational zeros. Candidates may fail. Testing confirms actual roots.
Start with the constant term. List its factors. Then inspect the leading coefficient. List its factors. Combine constant factors over leading factors. Reduce fractions. The set contains every rational zero.
Read the Candidate Rule
For a polynomial anxn + … + a1x + a0, each rational zero has form p/q. Here p divides a0. Here q divides an. Signs may be positive or negative. A candidate must be written in lowest terms. This avoids duplicates.
Consider 2x3 − 3x2 − 8x + 12. Factors come from 12 and 2. Candidates include ±1, ±2, ±3, ±4, ±6, ±12, and halves. Testing narrows. A zero of 2 proves x − 2 is a factor.
Test With Synthetic Division
Synthetic division gives a fast verification method. Place the candidate beside coefficients. Bring down the first term. Multiply it by the candidate. Add the result to the next coefficient. Repeat to the remainder.
A remainder of zero confirms a rational zero. The new coefficient row forms a lower-degree quotient. You can test the quotient again. Repeated testing finds additional rational roots. It also shows repeated roots clearly. A nonzero remainder rejects it.
Use the Calculator Results
Enter the degree and every coefficient, including zeros between powers. The calculator generates reduced candidates automatically. It evaluates each candidate using Horner’s method. The result table marks verified zeros. It also shows residual values and synthetic-division quotients.
The factor summary builds from confirmed zeros. A remaining polynomial may still have irrational or complex roots. In that case, the theorem has done its job. It found every rational possibility. Use graphing, formulas, or numerical methods afterward as needed.
Avoid Common Input Errors
Do not skip missing powers. For x4 − 5x2 + 4, enter 1, 0, −5, 0, 4. Incorrect order changes the polynomial. Keep coefficients as integers. Decimal coefficients need conversion before applying the standard theorem.
Remember that zero deserves special attention. When the constant term is zero, x is a factor. Divide by x first. Then apply the theorem to the smaller polynomial. This calculator performs that process while checking the remaining candidates carefully.
Rational Zero Theorem FAQs
1. What does the Rational Zero Theorem find?
It lists every possible rational root of a polynomial with integer coefficients. You must still test each possibility. A listed candidate may not actually make the polynomial equal zero.
2. What does p/q represent?
The numerator p is a factor of the constant term. The denominator q is a factor of the leading coefficient. Include both positive and negative versions.
3. Does every candidate become a root?
No. The theorem creates a complete candidate list, not a guaranteed root list. Direct substitution or synthetic division decides whether each candidate is valid.
4. Why are fractions reduced?
Reducing fractions removes duplicates. For example, 2/4 and 1/2 describe the same number. Testing each value once keeps the result table accurate and easier to read.
5. Can the constant term be zero?
Yes. A zero constant means x is a factor, so zero is a root. Divide by x first, then use the theorem on the quotient to find further rational roots.
6. Can I use decimal coefficients?
The standard theorem needs integer coefficients. Convert decimal coefficients to integers by multiplying the whole polynomial by a suitable power of ten before using the calculator.
7. What is synthetic division used for?
Synthetic division tests a candidate efficiently. A remainder of zero verifies the root. The resulting quotient lowers the degree and helps continue the factor search.
8. What if there are no verified rational zeros?
The polynomial can still have real or complex roots. They may be irrational. Use graphing, the quadratic formula, numerical methods, or other algebraic tools next.
9. Can a polynomial have repeated rational zeros?
Yes. Test the quotient after every successful division. If the same candidate again gives remainder zero, that root has multiplicity greater than one.
10. Why does the graph help?
The plot shows where the polynomial crosses or touches the x-axis. It supports the table results and can reveal whether unverified candidates are far from roots.
11. What does the CSV export include?
The CSV contains the polynomial, factor lists, every tested candidate, residual values, verification status, and synthetic-division quotients for confirmed rational zeros.