Question

graph the function \\(f(x) = x^3 - 3x - 2\\). based on the graph, which value for \\(x\\) is a double root of this function?

-2
-1
1
2

Explanation:

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<reasoning>

Find the roots of the cubic function

\[

$$\begin{aligned} f(x) &= x^3 - 3x - 2 \\ &= x^3 + x^2 - x^2 - x - 2x - 2 \\ &= x^2(x + 1) - x(x + 1) - 2(x + 1) \\ &= (x + 1)(x^2 - x - 2) \end{aligned}$$

\]

Factor the remaining quadratic expression

\[

$$\begin{aligned} x^2 - x - 2 &= (x - 2)(x + 1) \\ f(x) &= (x + 1)^2(x - 2) \end{aligned}$$

\]

Identify the double root

\[

$$\begin{aligned} &\text{Roots: } x = -1 \text{ (multiplicity 2), } x = 2 \text{ (multiplicity 1)} \\ &\text{Double root: } x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>-2</mcq-option>
<mcq-correct>-1</mcq-correct>
<mcq-option>1</mcq-option>
<mcq-option>2</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "xxx - 3*x - 2",
"latex": "x^3 - 3x - 2"
},
-3,
3
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
},
{
"type": "point",
"params": [
[-1, 0]
],
"properties": {
"name": "(-1, 0) Double Root",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [
[2, 0]
],
"properties": {
"name": "(2, 0) Single Root",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
}
]
}
</plot>

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"question_type": "Multiple Choice",
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"Mathematics",
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"Polynomial Roots"
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Answer:

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<reasoning>

Find the roots of the cubic function

\[

$$\begin{aligned} f(x) &= x^3 - 3x - 2 \\ &= x^3 + x^2 - x^2 - x - 2x - 2 \\ &= x^2(x + 1) - x(x + 1) - 2(x + 1) \\ &= (x + 1)(x^2 - x - 2) \end{aligned}$$

\]

Factor the remaining quadratic expression

\[

$$\begin{aligned} x^2 - x - 2 &= (x - 2)(x + 1) \\ f(x) &= (x + 1)^2(x - 2) \end{aligned}$$

\]

Identify the double root

\[

$$\begin{aligned} &\text{Roots: } x = -1 \text{ (multiplicity 2), } x = 2 \text{ (multiplicity 1)} \\ &\text{Double root: } x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>-2</mcq-option>
<mcq-correct>-1</mcq-correct>
<mcq-option>1</mcq-option>
<mcq-option>2</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "xxx - 3*x - 2",
"latex": "x^3 - 3x - 2"
},
-3,
3
],
"properties": {
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{
"type": "point",
"params": [
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"size": 4,
"color": "#5583F2",
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}
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</plot>

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"Polynomial Roots"
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