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

Jan 12, 2016

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:

$f \left(- 3\right) = {\left(- 3\right)}^{3} + 5 \cdot {\left(- 3\right)}^{2} + 7 \cdot \left(- 3\right) + 3 = - 27 + 45 - 21 + 3 = 18 - 18 = 0$ and

$f \left(- 1\right) = {\left(- 1\right)}^{3} + 5 \cdot {\left(- 1\right)}^{2} + 7 \cdot \left(- 1\right) + 3 = - 1 + 5 - 7 + 3 = 4 - 4 = 0$.

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]}