Question

How do you use the rational root theorem to list all possible roots for `2x^3+5x^2+4x+1=0`?

Answer 1

The "possible" rational roots are: `+-1/2, +-1`

The actual roots are: `-1/2, -1, -1`

Explanation:

Given:

`f(x) = 2x^3+5x^2+4x+1`

Note that the rational root theorem will only tell us about rational zeros, not necessarily all zeros.

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `1` and `q` a divisor of the coefficient `2` of the leading term.

That means that the only possible rational zeros are:

`+-1/2`, `+-1`

In this particular example note that all of the coefficients are positive. Hence `f(x)` has no positive zeros, so let's check `f(-1)` first:

`f(-1) = -2+5-4+1 = 0`

So `x=-1` is a zero and `(x+1)` a factor:

`2x^3+5x^2+4x+1 = (x+1)(2x^2+3x+1)`

Note that `-1` is also a zero of `2x^2+3x+1`, since `2-3+1 = 0`:

`2x^2+3x+1 = (x+1)(2x+1)`

So the third zero is `x=-1/2`

So in this particular example all of the roots were rational and therefore findable with the help of the rational roots theorem.