Final Answer:
The Rational Zero Theorem identifies potential rational roots as factors of the constant term (here, 6) divided by factors of the leading coefficient (here, 1). Possible rational roots for the function f(x)=x³ +7x²-13x+6 are ±1, ±2, ±3, and ±6.
Step-by-step explanation:
The Rational Zero Theorem serves as a guideline to determine potential roots for a polynomial equation. In this case, for the polynomial f(x)=x³ +7x²-13x+6, the theorem suggests that any rational root must be a factor of the constant term (here, 6) divided by a factor of the leading coefficient (here, 1). This leads to a set of possible rational roots: ±1, ±2, ±3, and ±6.
These potential roots are derived by considering all the possible combinations of factors of the constant term (6) divided by the factors of the leading coefficient (1). By calculating the factors of 6 (1, 2, 3, and 6) and 1 (only 1 itself), the Rational Zero Theorem offers a range of potential rational roots for the polynomial equation.
However, it's crucial to note that while these are potential rational roots based on the theorem, the theorem doesn't guarantee that any of these values will be actual roots of the polynomial equation. Additional methods such as synthetic division or other root-finding techniques are often necessary to determine the actual roots or factors of the polynomial equation.
The Rational Zero Theorem acts as a valuable tool in narrowing down the potential rational roots of a polynomial equation, aiding in the process of identifying possible solutions. However, it doesn't provide the definitive roots; instead, it assists in reducing the search space for potential solutions.