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.
1 Answer | Add Yours
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).
Find the remaining zeros using the quadratic formula:
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))`
Join to answer this question
Join a community of thousands of dedicated teachers and students.Join eNotes