for each of the following find all rational roots of the polynomial equation 2x cube - 5 x square + 1=0

Given:

The equation of polynomial is

To find:

All rational roots of the polynomial.

Solution:

According to the rational root theorem, all the possible rational roots of a polynomial are defined as

where, p is a factor of constant term and q is factor of leading coefficient.

We have,

Here, leading coefficient is 2 and constant term is 1.

Factors of 1 are ±1.

Factors of 2 are ±1, ±2.

Using rational root theorem, we get

Therefore, all possible rational roots of the given polynomial are .

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1