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# f(x)= 3x^3 -23x^2 -29x -7 List all possible (or potential) rational zeros for the...

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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.

Posted by cassmith111 on February 14, 2012 at 7:57 AM via web and tagged with correct answer, help needed, math, show all work

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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))`

Posted by taylormath on February 14, 2012 at 10:57 AM (Answer #1)