To determine which system of equations can be used to find the roots of the equation [tex]\( 4x^5 - 12x^4 + 6x = 5x^3 - 2x \)[/tex], let’s start by rewriting the given equation in a standard form, with all terms on one side of the equality:
[tex]\[ 4x^5 - 12x^4 + 6x - (5x^3 - 2x) = 0 \][/tex]
First, distribute the negative sign:
[tex]\[ 4x^5 - 12x^4 + 6x - 5x^3 + 2x = 0 \][/tex]
Now combine like terms:
[tex]\[ 4x^5 - 12x^4 - 5x^3 + 8x = 0 \][/tex]
From this equation, we need to identify a system of two equations that represent this polynomial. To verify this, we compare each provided system of equations to see which one correctly equates our polynomial.
1. [tex]\(\left\{\begin{array}{l}y = -4x^5 + 12x^4 - 6x \\ y = 5x^3 - 2x\end{array}\right.\)[/tex]
This system forms:
[tex]\[ -4x^5 + 12x^4 - 6x = 5x^3 - 2x \][/tex]
This does not match our polynomial [tex]\(4x^5 - 12x^4 - 5x^3 + 8x = 0\)[/tex]. So, this is incorrect.
2. [tex]\(\left\{\begin{array}{l}y = 4x^5 - 12x^4 - 5x^3 + 8x \\ y = 0\end{array}\right.\)[/tex]
This system is:
[tex]\[ 4x^5 - 12x^4 - 5x^3 + 8x = 0 \][/tex]
This matches exactly with our polynomial. This is a valid representation.
3. [tex]\(\left\{\begin{array}{l}y = 4x^5 - 12x^4 + 8x \\ y = -5x^3 + 2x\end{array}\right.\)[/tex]
This system forms:
[tex]\[ 4x^5 - 12x^4 + 8x = -5x^3 + 2x \][/tex]
This does not match our polynomial [tex]\(4x^5 - 12x^4 - 5x^3 + 8x = 0\)[/tex]. So, this is also incorrect.
4. [tex]\(\left\{\begin{array}{l}y = 4x^5 - 12x^4 + 6x \\ y = 5x^3 - 2x\end{array}\right.\)[/tex]
This system forms:
[tex]\[ 4x^5 - 12x^4 + 6x = 5x^3 - 2x \][/tex]
This does not match our polynomial [tex]\(4x^5 - 12x^4 - 5x^3 + 8x = 0\)[/tex]. So, this is incorrect as well.
The only correct system of equations that can be used to find the roots of the equation is:
[tex]\[ \left\{\begin{array}{l}y = 4x^5 - 12x^4 - 5x^3 + 8x \\ y = 0\end{array}\right. \][/tex]
Thus, the correct choice is:
[tex]\[
\left\{\begin{array}{l}y = 4x^5 - 12x^4 - 5x^3 + 8x \\ y = 0\end{array}\right.
\][/tex]
The index of this system in our list of choices is 4.
