Question 1029993
your equation is:


y = 7x^3 -x^2 + 2


the rational roots theorem states:


if f(x) has integer coefficients and p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient.


the problem states that IF p/q is a rational zero.


p/q has to be in simplified form, based on the definition.


to find the possible rational zeroes, you would take all the factors of p and all the factors of q and form all possible fractions of p/q from them.


in your equation of 7x^3 - x^1 + 2, you get:


p = 2 and q = 7.


the factors of 2 are 1 and 2.
the factors of 7 are 1 and 7.


the possible factors of p/q are 2/7, 2/1, 1/7, 1/1.


this results in possible factors or 1, 2, 1/7, 2/7.


you would then use synthetic division to find out if any of these are solutions to the equation.


you will find that none of them are.


if you graph the equation of 7x^3 - x^2 + 2, you will find that it has one zero at -.6142924033.


this is not a rational number as far as i can tell.


translate all your possible rational zeroes to decimals and you will find that they are equal to plus or minus 1, 2, .1428571429, .2857142857.


no cigars.


i did all the possible combinations and came up with no rational zeroes.


my worksheet is attached for you to review if you wish.


<img src = "http://theo.x10hosting.com/2016/041513.jpg" alt="$$$" </>


here's a video on youtube that describe the procedure used by the author in finding the rational roots if they exist.
he goes one step further and finds all the roots.


<a href = "https://www.youtube.com/watch?v=hJIgO0T5Wi8" target = "_blank">https://www.youtube.com/watch?v=hJIgO0T5Wi8</a>


he lists the possible rational roots.
then he uses synthetic division to narrow down the selections.
he gets down to the point where he has a quadratic equation left and he then solves for the remaining roots of the quadratic.


his roots for the equation becomes:


x = -3 or x = 1 or x = plus or minus sqrt(2).


the following graph shows the equation he factored using the rational roots theorem and the roots and the factors that are formed from those roots.


<img src = "http://theo.x10hosting.com/2016/041514.jpg" alt="$$$" </>