Answer :
To determine the [tex]\( x \)[/tex]-intercepts of the graph of the polynomial [tex]\( y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. That is, we need to solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 = 0. \][/tex]
By solving this polynomial equation for [tex]\( x \)[/tex], we obtain the [tex]\( x \)[/tex]-intercepts. These are the points where the graph of the function crosses the [tex]\( x \)[/tex]-axis.
The roots (solutions) of this polynomial equation are:
[tex]\[ x = -3 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts of the graph are the points where [tex]\( x \)[/tex] equals these values and [tex]\( y \)[/tex] is zero. We can write these intercepts as:
[tex]\[ (-3, 0), (-1, 0), (2, 0) \][/tex]
When comparing these points with the provided answer choices, we find that the corresponding choice is:
B. [tex]\((-3,0),(-1,0),(2,0)\)[/tex]
Therefore, the correct answer is:
[tex]\[ B. (-3,0),(-1,0),(2,0) \][/tex]
[tex]\[ -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 = 0. \][/tex]
By solving this polynomial equation for [tex]\( x \)[/tex], we obtain the [tex]\( x \)[/tex]-intercepts. These are the points where the graph of the function crosses the [tex]\( x \)[/tex]-axis.
The roots (solutions) of this polynomial equation are:
[tex]\[ x = -3 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts of the graph are the points where [tex]\( x \)[/tex] equals these values and [tex]\( y \)[/tex] is zero. We can write these intercepts as:
[tex]\[ (-3, 0), (-1, 0), (2, 0) \][/tex]
When comparing these points with the provided answer choices, we find that the corresponding choice is:
B. [tex]\((-3,0),(-1,0),(2,0)\)[/tex]
Therefore, the correct answer is:
[tex]\[ B. (-3,0),(-1,0),(2,0) \][/tex]
