Answer:
±1, ±5, ±7, and ±35.
Step-by-step explanation:
The possible zeros of a polynomial are the values of x that make the polynomial equal to zero. To find the possible zeros of the polynomial f(x) = x^3 - 5x^2 - 7x + 35, we can use the Rational Zero Theorem.
The Rational Zero Theorem states that if a polynomial has rational zeros, then they must be of the form p/q, where p is a factor of the constant term (in this case, 35) and q is a factor of the leading coefficient (in this case, 1).
The factors of 35 are ±1, ±5, ±7, and ±35. The factors of 1 are ±1. Therefore, the possible rational zeros are ±1, ±5, ±7, and ±35.
To find the actual zeros, we can use synthetic division or substitution. By substituting each possible zero into the polynomial, we can determine if it results in f(x) = 0.
For example, let's substitute x = 1 into f(x):
f(1) = (1)^3 - 5(1)^2 - 7(1) + 35
= 1 - 5 - 7 + 35
= 24
Since f(1) is not equal to zero, x = 1 is not a zero of the polynomial. We can repeat this process for each possible zero to determine the actual zeros.
To summarize, the possible zeros of f(x) = x^3 - 5x^2 - 7x + 35 are ±1, ±5, ±7, and ±35. However, to find the actual zeros, we need to substitute each possible zero into the polynomial and determine which values make f(x) equal to zero.
