Final answer:
To find the rational zeros of a function, you can use the Rational Root Theorem. By testing the factors of the constant term divided by factors of the leading coefficient, you can determine which ones are zeros of the function. The rational zeros for the given function are -1, 3, -7, 7, -9, 21, and -63.
Step-by-step explanation:
To find the rational zeros of the function f(x) = x^3 - x^2 - 65x - 63, you can use the Rational Root Theorem. According to the theorem, any rational zero of the function will be a factor of the constant term (-63) divided by a factor of the leading coefficient (1). The factors of -63 are: 1, -1, 3, -3, 7, -7, 9, -9, 21, -21, 63, and -63. The factors of 1 are: 1 and -1. By testing these factors, you can determine which ones are zeros of the function.
For example, if you test x = 1, f(1) = 1^3 - 1^2 - 65(1) - 63 = -128. Since f(1) is not equal to zero, x = 1 is not a zero of the function. By testing all the factors, you can find the rational zeros of the function.
The rational zeros for the function f(x) = x^3 - x^2 - 65x - 63 are:
- x = -1
- x = 3
- x = -7
- x = 7
- x = -9
- x = 21
- x = -63