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

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

Question

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

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

Accepted Answer

{ "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" ] } ## 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 $(0, 64)$. Replace the values b, c, and d to write function f. $f(x) =$ $(x-1)(x+1)^2(x-4)^3$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Multiplicity of Zeros" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the zeros and their multiplicities

Substitute the values into the function template

Verify with the y-intercept

Answer: