Question

replace the values b, c, and d to write function f.

\\f(x) = (x-b)(x-c)^2(x-d)^3\\

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Zeros of Polynomials"
],
"new_concepts": [
"Multiplicity of Zeros",
"Polynomial Graph Analysis"
],
"current_concepts": [
"Zeros of Polynomials",
"Multiplicity of Zeros",
"Polynomial Graph Analysis"
]
}
</pre_analysis>

<reasoning>

Identify the zeros and their multiplicities

Using the Zeros of Polynomials knowledge point
\[

$$\begin{aligned} &\text{At } x = 1, \text{ the graph crosses the } x\text{-axis linearly (multiplicity 1): } b = 1\\ &\text{At } x = -1, \text{ the graph is tangent to and bounces off the } x\text{-axis (multiplicity 2): } c = -1\\ &\text{At } x = 4, \text{ the graph has an inflection point crossing the } x\text{-axis (multiplicity 3): } d = 4 \end{aligned}$$

\]

Substitute the values into the function template

\[

$$\begin{aligned} f(x) &= (x - 1)(x - (-1))^2(x - 4)^3\\ f(x) &= (x - 1)(x + 1)^2(x - 4)^3 \end{aligned}$$

\]

Verify with the y-intercept

\[

$$\begin{aligned} f(0) &= (0 - 1)(0 + 1)^2(0 - 4)^3\\ f(0) &= (-1)(1)(-64) = 64 \end{aligned}$$

\]
The calculated \(y\)-intercept matches the graph's \(y\)-intercept at \((0, 64)\).
</reasoning>

<answer>
Replace the values b, c, and d to write function f.
\(f(x) =\) <blank>\((x-1)(x+1)^2(x-4)^3\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Multiplicity of Zeros"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Zeros of Polynomials"
],
"new_concepts": [
"Multiplicity of Zeros",
"Polynomial Graph Analysis"
],
"current_concepts": [
"Zeros of Polynomials",
"Multiplicity of Zeros",
"Polynomial Graph Analysis"
]
}
</pre_analysis>

<reasoning>

Identify the zeros and their multiplicities

Using the Zeros of Polynomials knowledge point
\[

$$\begin{aligned} &\text{At } x = 1, \text{ the graph crosses the } x\text{-axis linearly (multiplicity 1): } b = 1\\ &\text{At } x = -1, \text{ the graph is tangent to and bounces off the } x\text{-axis (multiplicity 2): } c = -1\\ &\text{At } x = 4, \text{ the graph has an inflection point crossing the } x\text{-axis (multiplicity 3): } d = 4 \end{aligned}$$

\]

Substitute the values into the function template

\[

$$\begin{aligned} f(x) &= (x - 1)(x - (-1))^2(x - 4)^3\\ f(x) &= (x - 1)(x + 1)^2(x - 4)^3 \end{aligned}$$

\]

Verify with the y-intercept

\[

$$\begin{aligned} f(0) &= (0 - 1)(0 + 1)^2(0 - 4)^3\\ f(0) &= (-1)(1)(-64) = 64 \end{aligned}$$

\]
The calculated \(y\)-intercept matches the graph's \(y\)-intercept at \((0, 64)\).
</reasoning>

<answer>
Replace the values b, c, and d to write function f.
\(f(x) =\) <blank>\((x-1)(x+1)^2(x-4)^3\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Multiplicity of Zeros"
]
}
</post_analysis>