To determine which system of equations can be used to find the roots of the given equation [tex]\(12 x^3 - 5 x = 2 x^2 + x + 6\)[/tex], we need to follow a step-by-step approach to match the given equation to one of the systems.
First, we start by rearranging the given equation:
[tex]\[12 x^3 - 5 x = 2 x^2 + x + 6\][/tex]
We want to bring all terms to one side of the equation to set it equal to zero:
[tex]\[12 x^3 - 5 x - 2 x^2 - x - 6 = 0\][/tex]
Next, combine the like terms:
[tex]\[12 x^3 - 2 x^2 - 6 x - 6 = 0\][/tex]
This represents a single polynomial equation in [tex]\(x\)[/tex] where we need to find the roots (values of [tex]\(x\)[/tex] that satisfy the equation).
Now we need to determine which of the provided systems can represent this polynomial equation.
Let's analyze each system:
1. [tex]\(\left\{ \begin{array}{l} y = 12x^3 - 5x \\ y = 2x^2 + x + 6 \end{array} \right.\)[/tex]
- Combining the two together: [tex]\(12x^3 - 5x = 2x^2 + x + 6\)[/tex]
- This is exactly the original form of the given polynomial equation before rearranging.
2. [tex]\(\left\{ \begin{array}{l} y = 12x^3 - 5x + 6 \\ y = 2x^2 + x \end{array} \right.\)[/tex]
- Combining the two together: [tex]\(12x^3 - 5x + 6 = 2x^2 + x\)[/tex]
- This form does not directly match our polynomial equation after reorganizing.
3. [tex]\(\left\{ \begin{array}{l} y = 12x^3 - 2x^2 - 6x \\ y = 6 \end{array} \right.\)[/tex]
- Combining the two together: [tex]\(12x^3 - 2x^2 - 6x = 6\)[/tex]
- This form does not match our polynomial equation because it simplifies incorrectly.
4. [tex]\(\left\{ \begin{array}{l} y = 12x^3 - 2x^2 - 6x - 6 \\ y = 0 \end{array} \right.\)[/tex]
- Combining the two together: [tex]\(12x^3 - 2x^2 - 6x - 6 = 0\)[/tex]
- This matches the form of our rearranged polynomial equation exactly.
Thus, the system of equations that can be used to find the roots of the given polynomial equation is:
[tex]\[
\boxed{\left\{\begin{array}{l}
y = 12x^3 - 2x^2 - 6x - 6 \\
y = 0
\end{array}\right.}
\][/tex]
