Question

the following equation is given. complete parts (a)-(c).
\\x^3 - 5x^2 - 4x + 20 = 0\\

a. list all rational roots that are possible according to the rational zero theorem.

1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20
(use a comma to separate answers as needed.)

Explanation:

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<reasoning>

Identify the coefficients of the polynomial

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &a_n = 1\\ &a_0 = 20 \end{aligned}$$

\]

Find the factors of the constant term and leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &p \in \{\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\}\\ &q \in \{\pm 1\} \end{aligned}$$

\]

Determine the possible rational roots

Using the Rational Root Theorem knowledge point
\[
\frac{p}{q} \in \{\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\}
\]
</reasoning>

<answer>
The following equation is given. Complete parts (a)–(c).
\[x^3 - 5x^2 - 4x + 20 = 0\]
a. List all rational roots that are possible according to the Rational Zero Theorem.
<blank>\(1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20\)</blank>
(Use a comma to separate answers as needed.)
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Algebra",
"Rational Root Theorem"
]
}
</post_analysis>

Answer:

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<reasoning>

Identify the coefficients of the polynomial

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &a_n = 1\\ &a_0 = 20 \end{aligned}$$

\]

Find the factors of the constant term and leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &p \in \{\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\}\\ &q \in \{\pm 1\} \end{aligned}$$

\]

Determine the possible rational roots

Using the Rational Root Theorem knowledge point
\[
\frac{p}{q} \in \{\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\}
\]
</reasoning>

<answer>
The following equation is given. Complete parts (a)–(c).
\[x^3 - 5x^2 - 4x + 20 = 0\]
a. List all rational roots that are possible according to the Rational Zero Theorem.
<blank>\(1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20\)</blank>
(Use a comma to separate answers as needed.)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rational Root Theorem"
]
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