Question

How do you find all the real and complex roots of `f(x) = x^3 + 5x^2 + 7x + 3`?

Answer 1

The roots are `x=-3, -1` (the second is a double-root ). The quickest way to find them is to use your calculator to help you make guesses and then check your guesses by substitution.

Explanation:

The graph is shown further below. It can help you guess the answers. Then we can check them as follows:

`f(-3)=(-3)^3+5*(-3)^2+7*(-3)+3=-27+45-21+3=18-18=0` and

`f(-1)=(-1)^3+5*(-1)^2+7*(-1)+3=-1+5-7+3=4-4=0`.

That `-1` is a double root can be checked via synthetic division (or polynomial long division). You can check the details. It can be guessed since the graph seems to just "touch" the `x` axis at `x=-1` rather than "passing through" the `x`-axis at that point.

If you didn't have technology to help you with all this, the old-school way of doing it is to use the Rational Roots Theorem to help you in your guessing.

The fact that there are only 3 roots (counting multiplicities), and non more, is a consequence of the famed Fundamental Theorem of Algebra .

graph{x^3+5x^2+7x+3 [-5.48, 4.52, -1.94, 3.06]}