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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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cassmith111 | Honors

Posted February 14, 2012 at 7:57 AM via web

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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.

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taylormath | High School Teacher | (Level 2) Adjunct Educator

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

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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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