QUESTION IMAGE
Question
an incomplete graph of the polynomial function \\(f(x) = -x^3 - 2x^2 + 13x - 10\\) is shown on the right.
a. find all zeros of the function.
b. without using a graphing utility, draw a complete graph of the function.
a. the zeros are
(use a comma to separate answers as needed.)
b. choose the correct graph of the function below. the scale for each graph is \\(-10, 10, 1\\) by \\(-80, 45, 5\\).
<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Rational Root Theorem",
"Synthetic Division",
"Quadratic Formula",
"End Behavior of Polynomials",
"Y-Intercept of Polynomial"
],
"new_concepts": [],
"current_concepts": [
"Rational Root Theorem",
"Synthetic Division",
"Quadratic Formula",
"End Behavior of Polynomials",
"Y-Intercept of Polynomial",
"Finding Zeros of Factored Polynomials"
]
}
</pre_analysis>
<reasoning>
Find rational zeros
Using the Rational Root Theorem knowledge point
\[
\text{Possible rational roots of } f(x) = -x^3 - 2x^2 + 13x - 10 \text{ are factors of } -10: \pm 1, \pm 2, \pm 5, \pm 10.
\]
\[
f(1) = -(1)^3 - 2(1)^2 + 13(1) - 10 = -1 - 2 + 13 - 10 = 0 \implies x = 1 \text{ is a zero.}
\]
Divide to find remaining factors
Using the Synthetic Division knowledge point
\[
\]
\[
f(x) = (x - 1)(-x^2 - 3x + 10) = -(x - 1)(x^2 + 3x - 10)
\]
Solve the quadratic factor
Using the Quadratic Formula knowledge point
\[
x^2 + 3x - 10 = 0 \implies (x + 5)(x - 2) = 0 \implies x = -5, \quad x = 2
\]
\[
\text{The complete set of zeros is } \{-5, 1, 2\}.
\]
Analyze end behavior and intercepts
Using the End Behavior of Polynomials and Y-Intercept of Polynomial knowledge points
\[
\text{Leading term: } -x^3 \implies \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = \infty
\]
\[
y\text{-intercept: } f(0) = -10
\]
\[
x\text{-intercepts: } (-5, 0), (1, 0), (2, 0)
\]
Match the correct graph
We look for a graph that:
- Starts high on the left (\(x \to -\infty, y \to \infty\)) and ends low on the right (\(x \to \infty, y \to -\infty\)).
- Crosses the \(x\)-axis at \(x = -5\), \(x = 1\), and \(x = 2\).
- Has a \(y\)-intercept at \((0, -10)\).
- Graph A: Starts low on the left, ends high on the right (incorrect orientation).
- Graph B: Starts high on the left, ends low on the right. Crosses at \(x = -5\), \(x = 1\), and \(x = 2\).
- Graph C: Starts low on the left, ends low on the right (incorrect end behavior).
- Graph D: Starts low on the left, ends high on the right (incorrect orientation).
Therefore, Graph B is the correct representation.
</reasoning>
<answer>
Question 1
The zeros are <blank>-5,1,2</blank>
Question 2
<mcq-option>(A) Graph starts low on the left, rises to the right</mcq-option>
<mcq-correct>(B) Graph starts high on the left, crosses at -5, 1, 2, and falls on the right</mcq-correct>
<mcq-option>(C) Graph starts low on both ends</mcq-option>
<mcq-option>(D) Graph starts low on the left, rises to the right with different intercepts</mcq-option>
</answer>
<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-xxx - 2xx + 13*x - 10",
"latex": "-x^3 - 2x^2 + 13x - 10"
},
-10,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "f(x)"
}
},
{
"type": "point",
"params": [[-5, 0]],
"properties": {
"name": "(-5,0)",
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 0]],
"properties": {
"name": "(1,0)",
"color": "#F2557F",
"wit…
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<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Rational Root Theorem",
"Synthetic Division",
"Quadratic Formula",
"End Behavior of Polynomials",
"Y-Intercept of Polynomial"
],
"new_concepts": [],
"current_concepts": [
"Rational Root Theorem",
"Synthetic Division",
"Quadratic Formula",
"End Behavior of Polynomials",
"Y-Intercept of Polynomial",
"Finding Zeros of Factored Polynomials"
]
}
</pre_analysis>
<reasoning>
Find rational zeros
Using the Rational Root Theorem knowledge point
\[
\text{Possible rational roots of } f(x) = -x^3 - 2x^2 + 13x - 10 \text{ are factors of } -10: \pm 1, \pm 2, \pm 5, \pm 10.
\]
\[
f(1) = -(1)^3 - 2(1)^2 + 13(1) - 10 = -1 - 2 + 13 - 10 = 0 \implies x = 1 \text{ is a zero.}
\]
Divide to find remaining factors
Using the Synthetic Division knowledge point
\[
\]
\[
f(x) = (x - 1)(-x^2 - 3x + 10) = -(x - 1)(x^2 + 3x - 10)
\]
Solve the quadratic factor
Using the Quadratic Formula knowledge point
\[
x^2 + 3x - 10 = 0 \implies (x + 5)(x - 2) = 0 \implies x = -5, \quad x = 2
\]
\[
\text{The complete set of zeros is } \{-5, 1, 2\}.
\]
Analyze end behavior and intercepts
Using the End Behavior of Polynomials and Y-Intercept of Polynomial knowledge points
\[
\text{Leading term: } -x^3 \implies \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = \infty
\]
\[
y\text{-intercept: } f(0) = -10
\]
\[
x\text{-intercepts: } (-5, 0), (1, 0), (2, 0)
\]
Match the correct graph
We look for a graph that:
- Starts high on the left (\(x \to -\infty, y \to \infty\)) and ends low on the right (\(x \to \infty, y \to -\infty\)).
- Crosses the \(x\)-axis at \(x = -5\), \(x = 1\), and \(x = 2\).
- Has a \(y\)-intercept at \((0, -10)\).
- Graph A: Starts low on the left, ends high on the right (incorrect orientation).
- Graph B: Starts high on the left, ends low on the right. Crosses at \(x = -5\), \(x = 1\), and \(x = 2\).
- Graph C: Starts low on the left, ends low on the right (incorrect end behavior).
- Graph D: Starts low on the left, ends high on the right (incorrect orientation).
Therefore, Graph B is the correct representation.
</reasoning>
<answer>
Question 1
The zeros are <blank>-5,1,2</blank>
Question 2
<mcq-option>(A) Graph starts low on the left, rises to the right</mcq-option>
<mcq-correct>(B) Graph starts high on the left, crosses at -5, 1, 2, and falls on the right</mcq-correct>
<mcq-option>(C) Graph starts low on both ends</mcq-option>
<mcq-option>(D) Graph starts low on the left, rises to the right with different intercepts</mcq-option>
</answer>
<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-xxx - 2xx + 13*x - 10",
"latex": "-x^3 - 2x^2 + 13x - 10"
},
-10,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "f(x)"
}
},
{
"type": "point",
"params": [[-5, 0]],
"properties": {
"name": "(-5,0)",
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 0]],
"properties": {
"name": "(1,0)",
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 0]],
"properties": {
"name": "(2,0)",
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, -10]],
"properties": {
"name": "(0,-10)",
"color": "#5583F2",
"withLabel": true
}
}
]
}
</plot>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"End Behavior of Polynomials"
]
}
</post_analysis>