Question

How do you find the number of roots for `f(x) = 3x^4 + x + 2` using the fundamental theorem of algebra?

Answer 1

By the FTOA, `f(x)` has exactly `4` Complex, possibly Real zeros counting multiplicity.

Further we find that none are Real.

Explanation:

The fundamental theorem of algebra (FTOA) tells us that any polynomial in one variable of degree `n > 0` has a Complex (possibly Real) zero.

A straightforward corollary of this, often stated as part of the FTOA is that a polynomial of degree `n > 0` has exactly `n` Complex, possibly Real, zeros counting multiplicity.

In our example:

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

is of degree `4` and therefore has `4` Complex, possibly Real, zeros counting multiplicity.

What else can we find out about these zeros?

The pattern of signs of the coefficients of `f(x)` is `+ + +`. With no changes of sign, we can deduce that `f(x)` has no positive Real zeros by Descartes' Rule of Signs.

The pattern of signs of `f(-x)` is `+ - +`. With `2` changes of sign, it means that `f(x)` has `0` or `2` negative Real zeros.

Note that when `abs(x) <= 1`, we have `3x^4 >= 0` and since `abs(x) < 2`, we find:

`f(x) > 0`

When `abs(x) > 1`, then `abs(3x^4) = abs(3x^3 * x) > 3 abs(x)`
Hence:

`f(x) > 0`

So `f(x)` has no Real zeros. All of its `4` zeros must be non-Real Complex.