Find the rational zeros for each polynomial funtion by using the rational zero theorem?

Always check for x = 1 first. If the coefficients add to 0 then x = 1 is a zero:

3 + 11 - 6 - 8 = 14 - 14 = 0

Use synthetic division to divide out the factor of (x - 1):

P(x) = (x - 1)(3x^2 + 14x + 8)

Trial and error the quadratic factor:

(3x )(x ) Same signs since sign in front of constant is + and sign in front of x-term is the sign to use:

(3x + )(x + ) Find factors of 8 placed properly that will make sum of inner and outer product equal 14x:

(3x + 2)(x + 4)

P(x) = (x - 1)(3x + 2)(x + 4)

It's zeros can be found by setting each linear factor equal to zero and solving:

x = - 4, - 2/3, 1

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Factors of - 8: +/- 1, 2, 4, 8

Factors of 3: +/- 1, 3

Possible rational roots: +/- 1/3, 1, 2, 2/3, 4, 4/3, 8, 8/3