Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual roots.
x^3 + 2x^2 + 3x + 6 = 0 (8 points)
To use the Rational Root Theorem to list all possible rational roots, you need to consider the factors of the constant term (in this case, 6) and the factors of the leading coefficient (in this case, 1).
The factors of 6 are ±1, ±2, ±3, and ±6. The factors of 1 are ±1.
According to the Rational Root Theorem, any rational root of the equation can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Therefore, the possible rational roots for the equation x^3 + 2x^2 + 3x + 6 = 0 are:
±1, ±2, ±3, and ±6.
To find the actual roots, you can use synthetic division or another root-finding method. By trying these factors as possible roots one by one, you can determine which (if any) are actual roots of the equation.
For example, let's try the possible root x = -1:
-1 │ 1 2 3 6
│ -1 -1 -2
--------------
1 1 2 4
The remainder is 4, not 0, so x = -1 is not a root.
By trying each of the possible rational roots, you can eventually find the actual roots of the equation.