I am not sure if I did this right, it is the theorem on bounds to establish integral bounds for roots of equation.

6x^3-7x^2+7x+9=0
1 6 -7 7 9 0 6 -7 7 9
6 -1 6 0 0 0
6 -1 6 15 6 -7 7 9

-1 6 -7 7 9
-6 13 20
6 -13 20 29

answer is -1<x<1

correct

I don't think so, better check it!

Does any know if I am correct? One person said yes, but someone else said no. There was not a reason posted why I was wrong.

Does any one know if I am correct? One person said yes, but some one said no. There was not a reason why or where I was wrong.

Bradley,

Hi presented your question to my online tutoring through my school and the tutor Arjun came back with the response that your answer is correct.

Hope this helps as I am currently having fun:) with this current subject in school as well.

Also to would be helpful if the person who thinks it was wrong would have responded with some suggestions of how to do it different.... but then they also responded with anonymous so there you go....

To establish integral bounds for the roots of the equation 6x^3 - 7x^2 + 7x + 9 = 0, you are using the theorem on bounds for roots of equations. This theorem helps determine a range or interval within which the roots of an equation lie.

To apply the theorem, you need to find the coefficients of the equation and arrange them in a table. In this case, the coefficients are 6, -7, 7, and 9. Begin by writing them in a row, followed by a 0 to separate the coefficients from the constant term.

1 6 -7 7 9 0

Next, carry out the necessary calculations. Start by finding the opposite of the first coefficient (-1 * 6 = -6). Write that in the second row below the first coefficient.

1 6 -7 7 9 0
-6

Now, add the second row to the first row, place the result beneath the second row, and continue with the next coefficient.

1 6 -7 7 9 0
-6
6 -1 6

Repeat these steps until you have covered all the coefficients, calculating and adding rows as needed:

1 6 -7 7 9 0
-6
6 -1 6
6 -13 20
6 -13 20 29

Once you have completed all the necessary calculations, examine the last row. In this case, the numbers are 6, -13, 20, 29. The last number in the last row is the constant term in the original equation.

The theorem states that the integral bounds for the roots of the equation lie between two consecutive sign changes in the last row. In this case, there is a sign change between -13 and 20 and between 20 and 29. Therefore, the integral bounds for the roots of the equation 6x^3 - 7x^2 + 7x + 9 = 0 are -1 < x < 1.