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Which system of equations can be used to find the roots of the equation [tex]$4x^5 - 12x^4 + 6x = 5x^3 - 2x$[/tex]?

A. [tex]\left\{\begin{array}{l}
y = -4x^5 + 12x^4 - 6x \\
y = 5x^3 - 2x
\end{array}\right.[/tex]

B. [tex]\left\{\begin{array}{l}
y = 4x^5 - 12x^4 - 5x^3 + 8x \\
y = 0
\end{array}\right.[/tex]

C. [tex]\left\{\begin{array}{l}
y = 4x^5 - 12x^4 + 8x \\
y = -5x^3 + 2x
\end{array}\right.[/tex]

D. [tex]\left\{\begin{array}{l}
y = 4x^5 - 12x^4 + 6x \\
y = 5x^3 - 2x
\end{array}\right.[/tex]

Sagot :

GinnyAnsweridn
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.
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