To find all possible rational zeros of a polynomial, the Rational Zeros Theorem (also known as the Rational Root Theorem) is used. Here are the steps:
- Arrange the polynomial in descending order of powers.
- Identify the constant term (the term without a variable) and list all of its factors, including both positive and negative.
- Identify the leading coefficient (the coefficient of the term with the highest power) and list all of its factors, including both positive and negative.
- Form all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. Simplify each fraction and include both positive and negative values of each.
- Test each candidate by substituting into the polynomial or using synthetic division to check if it produces zero.
This process generates a list of all possible rational zeros, though not all will necessarily be actual zeros of the polynomial. The actual zeros can be confirmed by substitution or division methods. For example, if the polynomial is 2x3+3x2−2x−62x^3+3x^2-2x-62x3+3x2−2x−6,
- The constant term is −6-6−6, the factors are ±1,±2,±3,±6\pm 1,\pm 2,\pm 3,\pm 6±1,±2,±3,±6.
- The leading coefficient is 222, the factors are ±1,±2\pm 1,\pm 2±1,±2.
- Possible rational zeros are all combinations of factors of −6-6−6 over factors of 222: ±1,±12,±2,±3,±32,±6\pm 1,\pm \frac{1}{2},\pm 2,\pm 3,\pm \frac{3}{2},\pm 6±1,±21,±2,±3,±23,±6.
Then each candidate is tested to find actual roots.
