Question

Identify all of the roots of the function.

Answer 1

The integer roots of the polynomial function $f(x)=x^3-5x^2-2x+24$ can be only among the divisors of free term.

The divisors of free term are:

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24.$

Check them:

1. $f(1)=1^3-5\cdot 1^2-2\cdot 1+24=18\neq 0$, 1 is not a root.

2. $f(-1)=(-1)^3-5\cdot (-1)^2-2\cdot (-1)+24=20\neq 0$, -1 is not a root.

3. $f(2)=2^3-5\cdot 2^2-2\cdot 2+24=8\neq 0$, 2 is not a root.

4. $f(-2)=(-2)^3-5\cdot (-2)^2-2\cdot (-2)+24=0$, -2 is a root.

5. $f(3)=3^3-5\cdot 3^2-2\cdot 3+24=0$, 3 is  a root.

6.  $f(-3)=(-3)^3-5\cdot (-3)^2-2\cdot (-3)+24=-42\neq 0$, -3 is not a root.

7. $f(4)=4^3-5\cdot 4^2-2\cdot 4+24=0$, 4 is  a root.

The cubic function has at most 3 roots, then

Answer: roots of the function are -2, 3 and 4.

Answer 2

F(x) = x3 - 5x2 - 2x + 24

roots -2, 3 and 4