$p=(w-30)\left(w^{2}+178 w+792\right.$ $\newline$ Given that $-89$ is a double zero of the polynomial equation, which of the following could be the graph of the equation in the $w p$ -plane? $\newline$ Choose $1$ answer: $\newline$ (A) $\newline$ (B) $\newline$ (c) $\newline$ $p$ $\newline$ Lesson $\newline$ tion graphs - Harder example $\newline$ aphs: medium $\newline$ is $\newline$ er nonlinear graphs: medium $\newline$ other nonlinear graphs | Lesson $\newline$ tors and graphs - Basic example

Full solution

p=(w-30)\left(w^{2}+178 w+792\right.

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Given that

-89

is a double zero of the polynomial equation, which of the following could be the graph of the equation in the

w p

-plane?

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Choose

1

answer:

\newline

(A)

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(B)

\newline

(c)

\newline

p

\newline

Lesson

\newline

tion graphs - Harder example

\newline

aphs: medium

\newline

\newline

er nonlinear graphs: medium

\newline

other nonlinear graphs | Lesson

\newline

tors and graphs - Basic example

Identify Given Information: Identify the given information and the implication of a double zero. A double zero means that the factor corresponding to that zero appears twice in the factorization of the polynomial.
Write Polynomial with Double Zero: Since $-89$ is a double zero, the polynomial can be written as $p=(w-30)(w+89)^2$ . This means that the factor $(w+89)$ is squared in the polynomial.
Expand to Find Third Zero: Expand the polynomial to find the third zero. We already have two zeros at $w=-89$ (double zero), and we need to find the third zero from the factor $(w-30)$ .
Solve for Third Zero: Set the remaining factor equal to zero and solve for $w$ : $w - 30 = 0$ , which gives us $w = 30$ .
Determine All Zeros: Now we have all the zeros of the polynomial: $w = -89$ (with multiplicity $2$ ) and $w = 30$ . The graph of the polynomial in the $wp$ -plane will touch the $w$ -axis at $w = -89$ and cross the $w$ -axis at $w = 30$ .
Select Correct Graph: Choose the graph that shows the correct behavior at the zeros. The graph should touch the $w$ -axis at $w = -89$ and cross the $w$ -axis at $w = 30$ . Without the actual graphs labeled ( $A$ ), ( $B$ ), and ( $C$ ), we cannot visually select the correct graph. However, the description provided should match the correct graph.