The formula to calculate the rational zeros of a polynomial equation is:
\[ Z = \frac{\text{factors of constant term}}{\text{factors of leading coefficient}} \]
Where:
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 \( \frac{p}{q} \) 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.
Let's assume the following polynomial equation:
\[ 2x^3 - 3x^2 + 2x - 3 = 0 \]
Step 1: Find the factors of the constant term (-3):
\[ \text{Factors of -3:} \pm 1, \pm 3 \]
Step 2: Find the factors of the leading coefficient (2):
\[ \text{Factors of 2:} \pm 1, \pm 2 \]
Step 3: Form all possible ratios of these factors to find the rational zeros:
\[ \text{Rational Zeros:} \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2} \]
Therefore, the possible rational zeros of the polynomial are \( \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2} \).
