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.)

b. use synthetic division to test several possible rational roots in order to identify one actual root.

one rational root of the given equation is .
(simplify your answer.)

Explanation:

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<reasoning>

Identify the polynomial and possible rational roots

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &P(x) = x^3 - 5x^2 - 4x + 20 = 0\\ &\text{Constant term } a_0 = 20 \implies \text{Factors: } \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\\ &\text{Leading coefficient } a_n = 1 \implies \text{Factors: } \pm 1\\ &\text{Possible rational roots: } \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \end{aligned}$$

\]

Test possible roots using synthetic division

Using the Synthetic Division knowledge point
\[

$$\begin{aligned} &\text{Test } x = 1:\\ &\begin{array}{r|rrrr} 1 & 1 & -5 & -4 & 20 \\ & & 1 & -4 & -8 \\ \hline & 1 & -4 & -8 & 12 eq 0 \end{array}\\ &\text{Test } x = 2:\\ &\begin{array}{r|rrrr} 2 & 1 & -5 & -4 & 20 \\ & & 2 & -6 & -20 \\ \hline & 1 & -3 & -10 & 0 \end{array} \end{aligned}$$

\]

Confirm the actual rational root

Since the remainder is \(0\) when dividing by \(x - 2\), \(x = 2\) is a rational root of the equation.
</reasoning>

<answer>
One rational root of the given equation is <blank>2</blank>.
</answer>

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Answer:

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<reasoning>

Identify the polynomial and possible rational roots

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &P(x) = x^3 - 5x^2 - 4x + 20 = 0\\ &\text{Constant term } a_0 = 20 \implies \text{Factors: } \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\\ &\text{Leading coefficient } a_n = 1 \implies \text{Factors: } \pm 1\\ &\text{Possible rational roots: } \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \end{aligned}$$

\]

Test possible roots using synthetic division

Using the Synthetic Division knowledge point
\[

$$\begin{aligned} &\text{Test } x = 1:\\ &\begin{array}{r|rrrr} 1 & 1 & -5 & -4 & 20 \\ & & 1 & -4 & -8 \\ \hline & 1 & -4 & -8 & 12 eq 0 \end{array}\\ &\text{Test } x = 2:\\ &\begin{array}{r|rrrr} 2 & 1 & -5 & -4 & 20 \\ & & 2 & -6 & -20 \\ \hline & 1 & -3 & -10 & 0 \end{array} \end{aligned}$$

\]

Confirm the actual rational root

Since the remainder is \(0\) when dividing by \(x - 2\), \(x = 2\) is a rational root of the equation.
</reasoning>

<answer>
One rational root of the given equation is <blank>2</blank>.
</answer>

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