solve the following equation in the real number system

3x^3 - 20x^2 + 29x +12 = 0
Ok I thought I knew how to do this but I guess I don't. Help me! Please!

A general cubic is hard to solve. So, expect an easy root first. Any rational root of the form a/b will be such that a divides 12 and b divides 3

A little synthetic division will show that

3x^3 - 20x^2 + 29x +12
= (x-3)(3x^2 - 11x - 4)
= (x-3)(x-4)(3x+1)

SO, knowing the factors, you should have no trouble finding the roots . . .

Thank you that helps a lot!

To solve the equation 3x^3 - 20x^2 + 29x + 12 = 0, you can use a combination of factoring and the Rational Root Theorem.

Step 1: Look for Rational Roots
The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p is a factor of the constant term (12 in this case), and q is a factor of the leading coefficient (3 in this case).

In this equation, the possible rational roots p/q can be obtained by finding the factors of 12 (the constant term) and 3 (the leading coefficient). The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and the factors of 3 are ±1, ±3.

For simplicity, you can use synthetic division or a graphing calculator to test possible roots. By trying several values, you can quickly find the rational roots, if they exist.

Alternatively, you can use the rational root formula to narrow down the possible rational roots. The rational root formula states that if the equation has a rational root in the form of p/q, then p must be a factor of the constant term (12) and q must be a factor of the leading coefficient (3).

By applying the rational root formula, the possible rational roots can be:
±1/3, ±2/3, ±1, ±2, ±3, ±4, ±6, ±12.

Step 2: Test the Rational Roots
Using synthetic division or a graphing calculator, you can test the possible rational roots you obtained in step 1.

By testing these values, you will find that x = -4/3, x = -1 and x = 3 are the three rational roots of the equation.

Step 3: Factor the Equation
Since we have found the rational roots (-4/3, -1, and 3), we can factor them out. Divide the polynomial by each of the roots to get the factored form.

(x + 4/3)(x + 1)(x - 3) = 0

Step 4: Solve for x
Finally, set each factor equal to zero and solve for x:

x + 4/3 = 0 --> x = -4/3
x + 1 = 0 --> x = -1
x - 3 = 0 --> x = 3

Therefore, the real solutions to the equation 3x^3 - 20x^2 + 29x + 12 = 0 are x = -4/3, x = -1, and x = 3.