f(x)= 3x^3 -23x^2 -29x -7

** **List all possible (or potential) rational zeros for the polynomial above. Find all real zeros of the polynomial above and factor completely over the real numbers.

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Potential rational zeros for a polynomial are found by taking combinations of the factors of the last term and dividing by the factors of the first term. In the function f(x), the factors of 7 are 1 and 7. The factors of 3 are 1 and 3. A list of possible rational zeros are:

`+-1, +-7, +-1/3, +-7/3`

Using the list of possible rational zeros, the graph shows that -1/3 has good possibilities of being a rational zero.

Checking -1/3 using synthetic division does yield a zero remainder.

Therefore, -1/3 is a real rational zero. To find the remaining zeros, divide f(x) by the factor (3x+1).

`(3x^3-23x^2-29x-7)/(3x+1)=x^2-8x-7`

Find the remaining zeros using the quadratic formula:

`(8+-sqrt(64-4(1)(-7)))/(2(1))=(8+-sqrt(92))/2=4+-sqrt(23)`

The three roots are: **-1/3 , 4+sqrt(23), and 4-sqrt(23)**

The factors are : `(3x+1)(x-4-sqrt(23))(x-4+sqrt(23))`

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