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

1 Answer
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.


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


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]}