Solve 5x^3 – 6x^2 – 4x – 8 = 0
first try some factors of 8
after 2 attempt I found x = 2 works
so x-2 is a factor
by synthetic division ....
5x^3 – 6x^2 – 4x – 8
= (x-2)(5x^2 + 4x + 4)
now solve 5x^2 + 4x + 4 = 0
Use the formula, you will get two complex roots
To solve the equation 5x^3 – 6x^2 – 4x – 8 = 0, we'll proceed by using the Rational Root Theorem to find potential rational solutions.
Step 1: List all possible rational solutions.
According to the Rational Root Theorem, any rational solution of the form p/q must satisfy the following conditions:
- p is a factor of the constant term (in this case, it is 8).
- q is a factor of the leading coefficient (in this case, it is 5).
For p, the factors of 8 are ±1, ±2, ±4, and ±8.
For q, the factors of 5 are ±1 and ±5.
This gives us eight potential rational solutions:
±1/1, ±1/5, ±2/1, ±2/5, ±4/1, and ±4/5.
Step 2: Test the potential solutions using synthetic division.
We can test these potential solutions one by one using synthetic division until we find a solution that yields a remainder of 0.
To solve this particular equation, we can use numerical approximation methods or a graphing calculator to find the approximate solutions.