Use the rational zero theorem to list all possible rational zeros for the given function
f(x)=x^3-4x^2-19x-14
-2, -1, 7
try x = ±1, ±2, and ± 7
f(1) = 1 - 4 - 19 - 14 ≠ 0
f(-1) = -1 - 4 + 19 - 14 = 0
ahhh, so (x+1) is a factor
using algebraic division
x^3-4x^2-19x-14 = 0
(x+1)(x+2)(x-7) = 0
so x = -1, -2 or 7
To use the rational zero theorem, we need to find all possible rational zeros for the given function f(x) = x^3 - 4x^2 - 19x - 14.
The rational zero theorem states that if a rational number p/q is a zero of a polynomial function, then p must be a factor of the constant term (in this case, -14), and q must be a factor of the leading coefficient (in this case, 1).
So, to find all possible rational zeros, we need to list all the factors of -14 and divide them by all the factors of 1.
The factors of -14 are: -1, 1, -2, 2, -7, 7, -14, and 14.
The factors of 1 are: -1 and 1.
Dividing these factors, we get the following possible rational zeros:
-1/1 (or -1), 1/1 (or 1), -2/1 (or -2), 2/1 (or 2), -7/1 (or -7), 7/1 (or 7), -14/1 (or -14), and 14/1 (or 14).
Therefore, the possible rational zeros for the function f(x) = x^3 - 4x^2 - 19x - 14 are:
-1, 1, -2, 2, -7, 7, -14, and 14.