Question

How do you use the rational root theorem to find the roots of `3x^4 + 5x^3 + 3x^2 - 7x + 9 = 0`?

Answer 1

There are no rational roots.

Explanation:

Rational roots would be the factors of `9/3` that is 3. These can be 1,-1,3,-3. None of these values satisfy the given equation, hence it is concluded that there are no rational roots.

Answer 2

From the rational root theorem we find that the only possible rational roots are `+-1/3`, `+-1`, `+-3` and `+-9`.

None of these is a root, so the equation has no rational roots.

Explanation:

By the rational root theorem, any rational root of `3x^4+5x^3+3x^2-7x+9=0` must be of the form `p/q` where `p` and `q` are integers, `q != 0`, `p` a divisor of the constant term `9` and `q` a divisor of the coefficient `3` of the term `3x^4` of highest degree.

That means that the only possible rational roots are:

`+-1/3`, `+-1`, `+-3` and `+-9`

Let `f(x) = 3x^4+5x^3+3x^2-7x+9`

Then:

`f(1/3) = 65/9`

`f(-1/3) = 311/27`

`f(1) = 13`

`f(-1) = 17`

`f(3) = 393`

`f(-3) = 165`

`f(9) = 23517`

`f(-9) = 16353`

So none of the possible rational roots is a root of `f(x) = 0`

So `f(x) = 0` has no rational roots.

In fact it has no real roots at all:

graph{3x^4+5x^3+3x^2-7x+9 [-20, 20, -4, 16]}