Answer
x= 1, 3, -4
Work Step by Step
The Rational Zero Test can be used to find the rational zeros of the function.
The Rational Zero Test relies on finding possible zeros and testing to see if they are actually zeros. The possible zeros are all the factors of the constant term divided by all the factors of the leading coefficient.
f(x) = $x^{3}$ - 13x + 12
Factors of the constant: $\pm$ 1, $\pm$2, $\pm$3, $\pm$4, $\pm$6, $\pm$12
Factors of the leading coefficient: $\pm$1
All possible combinations:
x = $\frac{\pm 1}{\pm 1}$, $\frac{\pm 2}{\pm 1}$, $\frac{\pm 3}{\pm 1}$, $\frac{\pm 4}{\pm 1}$, $\frac{\pm 6}{\pm 1}$, $\frac{\pm 12}{\pm 1}$
x = 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12
Test the Possible Combinations:
x = 1:
f(1) = $1^{3}$ - 13(1) + 12
Since 0 = 0, x = 1 is a zero
x = 3:
f(3) = $3^{3}$ - 13(3) + 12
Since 0 = 0, x = 3 is a zero
x = -4:
f(1) = $(-4)^{3}$ - 13(-4) + 12
Since 0 = 0, x = -4 is a zero
