Question

how do you prove or disprove if -3 if a root of the function f(x)+x^3 + 5x^2 + 7x +6

Answer 1

A couple of ways come to mind:

1) evaluate f(-3). If it evaluates to 0, you've got a root at x = -3.
2) there's a theorem called the remainder theorem that says if you've got some polynomial P(x), and you divide it by (x-a), the remainder will be the value of the function at a. So, you could divide your function by (x-(-3) = (x+3) and see if the remainder is 0. If it is, there's a root at x = -3.
3) factor the polynomial and inspect the factors. If one of them is (x+3), you've got a root at x = -3
4) plug the equation into Wolfram|Alpha and see if it lists x = -3 among the roots :-)

Answer 2

ok thx