draw by hand a rough sketch of the graph of each function.

\p(x) = 5x^4 + 17x^3 + 6x^2 - 20x - 8\
\= (5x + 2)(x - 1)(x + 2)^2\

choose the correct graph below.

Question

draw by hand a rough sketch of the graph of each function.

\p(x) = 5x^4 + 17x^3 + 6x^2 - 20x - 8\
\= (5x + 2)(x - 1)(x + 2)^2\

choose the correct graph below.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Polynomial Zeros", "Polynomial Multiplicity", "End Behavior Analysis" ], "new_concepts": [], "current_concepts": [ "Polynomial Zeros", "Polynomial Multiplicity", "End Behavior Analysis", "Graphing Parabolas" ] } ## Analyze the end behavior of the polynomial Using the *End Behavior Analysis* knowledge point $$ \begin{aligned} &P(x) = 5x^4 + 17x^3 + 6x^2 - 20x - 8\ &\text{Degree } n = 4 \quad (\text{even})\ &\text{Leading coefficient } a_4 = 5 > 0\ &\lim_{x \to \infty} P(x) = \infty, \quad \lim_{x \to -\infty} P(x) = \infty \end{aligned} $$ ## Find the zeros and their multiplicities Using the *Polynomial Zeros* and *Polynomial Multiplicity* knowledge points $$ \begin{aligned} &P(x) = (5x + 2)(x - 1)(x + 2)^2 = 0\ &x = -\frac{2}{5} = -0.4 \quad (\text{multiplicity } 1 \implies \text{crosses } x\text{-axis})\ &x = 1 \quad (\text{multiplicity } 1 \implies \text{crosses } x\text{-axis})\ &x = -2 \quad (\text{multiplicity } 2 \implies \text{touches and turns around}) \end{aligned} $$ ## Determine the y-intercept Using the *Standard Form Polynomials* knowledge point $$ \begin{aligned} &P(0) = -8\ &\text{y-intercept is } (0, -8) \end{aligned} $$ ## Match with the correct graph option Using the *Graphing Parabolas* knowledge point $$ \begin{aligned} &\text{Graph must touch } x = -2 \text{ and turn back up.}\ &\text{Graph must cross } x = -0.4 \text{ and } x = 1.\ &\text{Graph must pass through } (0, -8).\ &\text{Option A shows a local minimum touching the } x\text{-axis at } x = -2,\ &\text{crossing at } x = -0.4 \text{ and } x = 1, \text{ and passing through } (0, -8). \end{aligned} $$

(A) Touches at $x = -2$, crosses at $x = -0.4$ and $x = 1$, y-intercept at $(0, -8)$

(B) Crosses at $x = -2$, touches at $x = 1$

(C) Touches at $x = 1$, crosses at $x = -2$

(D) Touches at $x = 0$, different end behavior

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QUESTION IMAGE

Question

Explanation:

Analyze the end behavior of the polynomial

Find the zeros and their multiplicities

Determine the y-intercept

Match with the correct graph option

Answer: