Enter all but one of the factors of the constant term and the factors of the leading coefficient into the calculator to determine the set of rational zeros; this calculator can also evaluate any of the variables given the others are known.
Rational Zeros Formula
The following formula is used to calculate the rational zeros of a polynomial equation:
Z = frac{{factors of constant term}}{{factors of leading coefficient}}Variables:
- Z is the set of rational zeros
- factors of constant term are the factors of the constant term of the polynomial equation
- factors of leading coefficient are the factors of the coefficient of the highest degree term of the polynomial equation
To calculate the rational zeros, find all the factors of the constant term and the factors of the leading coefficient. Then, divide the factors of the constant term by the factors of the leading coefficient to obtain the set of rational zeros.
What is a Rational Zeros?
A Rational Zero, also known as a Rational Root, is a concept in mathematics that is used to identify potential roots of a polynomial. The Rational Root Theorem states that if a polynomial has integer coefficients, then every rational zero will have the form p/q where p is a factor of the constant term and q is a factor of the leading coefficient. For example, if we have a polynomial equation like 2x^3 – 3x^2 + 2x – 3 = 0, the rational zeros of this polynomial can be found by listing all the possible factors of the constant term (-3) and the leading coefficient (2), and then forming all possible ratios of these factors. The rational zeros are the values that, when substituted into the polynomial, make the polynomial equal to zero. This theorem is a useful tool in finding the roots of a polynomial equation.
How to Calculate Rational Zeros?
The following steps outline how to find the rational zeros of a polynomial equation.
- First, write down the polynomial equation in standard form.
- Next, list all the possible rational zeros of the equation.
- Use the Rational Zero Theorem to narrow down the list of possible rational zeros.
- Divide the polynomial equation by each possible rational zero using synthetic division.
- If the remainder is zero, then the tested value is a rational zero.
- Repeat the process until all rational zeros have been found.
Example Problem:
Use the following polynomial equation as an example problem to test your knowledge.
Polynomial equation: 2x^3 – 5x^2 + 3x – 1 = 0
Possible rational zeros: ±1, ±1/2, ±1/3, ±1/6
Using synthetic division, test each possible rational zero to find the actual rational zeros.
