Answer:
To solve for the roots of the equation x³ + 3x² - 4 = 0, we can use the Rational Root Theorem and synthetic division.
Rational Root Theorem: The rational root theorem states that if a polynomial equation has integer coefficients, then any rational root of the equation must have a numerator that divides the constant term and a denominator that divides the leading coefficient.
In this case, the constant term is -4 and the leading coefficient is 1, so any rational root of the equation must be of the form x = p/q, where p is a factor of -4 and q is a factor of 1.
The possible rational roots of the equation are:
±1, ±2, ±4
Synthetic Division: Synthetic division can be used to test each of the possible rational roots and simplify the equation into a quadratic equation.
We can begin by testing x = 1:
1 3 -4
1|||__
1 4
1 4 0
The result of the division is x² + 4x + 0, which simplifies to x(x + 4). Therefore, one of the roots of the equation is x = -4.
We can then test x = -1:
1 3 -4
-1|||__
-1 -2
1 2 -6
The result of the division is x² + 2x - 6, which does not have any rational roots. Therefore, x = -1 is not a root of the equation.
Next, we can test x = 2:
1 3 -4
2|||__
2 10
1 5 6
The result of the division is x² + 5x + 6, which simplifies to (x + 2)(x + 3). Therefore, the other two roots of the equation are x = -2 and x = -3.
Therefore, the roots of the equation x³ + 3x² - 4 = 0 are x = -4, x = -2, and x = -3.