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The Rational Root Theorem is a very useful tool in algebra when dealing with polynomial functions. It helps us to identify possible rational zeros of a polynomial function. A rational zero is a possible solution to the polynomial equation set to zero, which is a fraction or a whole number. This theorem states that for a polynomial function like \( f(x) = x^3 - 2x^2 - 23x + 60 \), the rational zeros are formed using the factors of the constant term and the leading coefficient. In our example, the constant term is 60 and the leading coefficient is 1. Therefore, the possible rational zeros are the factors of 60, which include:

• ±1
• ±2
• ±3
• ±4
• ±5
• ±6
• ±10
• ±12
• ±15
• ±20
• ±30
• ±60

This theorem significantly limits the number of possible values to test, making it an efficient starting point in finding actual real zeros of a polynomial function.

Real zeros of a polynomial function are the x-values where the graph of the function crosses or touches the x-axis. They are the solutions to the equation \( f(x) = 0 \). In simpler terms, real zeros are the points where the polynomial equals zero.
Using the values identified by the Rational Root Theorem, we can test each to see if it is a zero of the polynomial. For example, if substituting \( x = -3 \) into \( f(x) \) results in zero, then \( x = -3 \) is a real zero. In our original problem, testing the possible zeros revealed that \( x = -3 \), \( x = 4 \), and \( x = 5 \) are indeed real zeros of the polynomial function \( f(x) = x^3 - 2x^2 - 23x + 60 \).
This means that at these values, the polynomial function will evaluate to zero, and thus the graph of this function would intersect the x-axis at these points.

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The individual terms take the form \( a_n x^n \) where \( a_n \) are coefficients and \( x^n \) represents the variable \( x \) raised to the nth power.
The function \( f(x) = x^3 - 2x^2 - 23x + 60 \) is a polynomial function of degree 3 because the highest power of \( x \) is 3. Each of the coefficients such as -2, -23, and the constant term 60 plays a role in shaping the graph of the polynomial.

• The degree of the polynomial determines the maximum number of real zeros it can have, which is the degree itself, 3 in this case.
• Polynomials can be simple or very complex, but this particular function exemplifies a cubic polynomial, where the degree equals 3.

Understanding the structure of polynomial functions aids in comprehending how they behave, especially when looking for zeros or graphing them.