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 2.
(simplify your answer.)

c. use the root from part (b) and solve the equation.

the solution set of \\x^3 - 5x^2 - 4x + 20 = 0\\ is \\{\quad\\}.
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Explanation:

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<reasoning>

List possible rational roots

Using the Rational Root Theorem knowledge point
\[

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

\]

Test possible roots using synthetic division

Using the Synthetic Division knowledge point
\[

$$\begin{aligned} &\text{Testing } x = 2:\\ &\begin{array}{r|rrrr} 2 & 1 & -5 & -4 & 20 \\ & & 2 & -6 & -20 \\ \hline & 1 & -3 & -10 & 0 \end{array}\\ &\text{Since the remainder is } 0\text{, } x = 2 \text{ is a root.} \end{aligned}$$

\]

Solve the remaining quadratic equation

Using the Solving by Radicals knowledge point
\[

$$\begin{aligned} &x^2 - 3x - 10 = 0\\ &(x - 5)(x + 2) = 0\\ &x = 5 \quad \text{or} \quad x = -2 \end{aligned}$$

\]

State the complete solution set

Combine all found roots to form the solution set:
\[
x = 2, \quad x = 5, \quad x = -2
\]
</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.)

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 <blank>\(2\)</blank>.
(Simplify your answer.)

c. Use the root from part (b) and solve the equation.
The solution set of \(x^3 - 5x^2 - 4x + 20 = 0\) is <blank>\(2, 5, -2\)</blank>.
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
</answer>

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Answer:

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<reasoning>

List possible rational roots

Using the Rational Root Theorem knowledge point
\[

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

\]

Test possible roots using synthetic division

Using the Synthetic Division knowledge point
\[

$$\begin{aligned} &\text{Testing } x = 2:\\ &\begin{array}{r|rrrr} 2 & 1 & -5 & -4 & 20 \\ & & 2 & -6 & -20 \\ \hline & 1 & -3 & -10 & 0 \end{array}\\ &\text{Since the remainder is } 0\text{, } x = 2 \text{ is a root.} \end{aligned}$$

\]

Solve the remaining quadratic equation

Using the Solving by Radicals knowledge point
\[

$$\begin{aligned} &x^2 - 3x - 10 = 0\\ &(x - 5)(x + 2) = 0\\ &x = 5 \quad \text{or} \quad x = -2 \end{aligned}$$

\]

State the complete solution set

Combine all found roots to form the solution set:
\[
x = 2, \quad x = 5, \quad x = -2
\]
</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.)

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 <blank>\(2\)</blank>.
(Simplify your answer.)

c. Use the root from part (b) and solve the equation.
The solution set of \(x^3 - 5x^2 - 4x + 20 = 0\) is <blank>\(2, 5, -2\)</blank>.
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
</answer>

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