Solve the equation [tex]x^3 + 3x^2 - 16x - 48 = 0[/tex].

According to the Rational Root Theorem, any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-48) and q is a factor of the leading coefficient.
So, the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.
By trying these values, we find that x = -3 is a root of the equation.

To find the other roots, we divide the equation by x + 3 using synthetic division.
Performing synthetic division, we get:
-3 | 1 3 -16 -48

| -3 0 48

_____________

1 0 -16 0
The result of the division is[tex]x^2 - 16 = 0[/tex].

We can now solve this quadratic equation by factoring or by using the quadratic formula.
Factoring the equation, we get:
(x - 4)(x + 4) = 0
Setting each factor equal to zero:
x - 4 = 0 or x + 4 = 0
Solving for x, we get:
x = 4 or x = -4

So, the solutions to the equation [tex]x^3 + 3x^2 - 16x - 48 = 0[/tex] are x = -3, x = 4, and x = -4.

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