Complete The Steps To Solve The Polynomial Equation $x^3 - 21x = -20$.According To The Rational Root Theorem, Which Number Is A Potential Root Of The Polynomial?A. -7 B. 0 C. 1 D. 3

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Understanding Polynomial Equations

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Polynomial equations are a fundamental concept in algebra, and solving them is a crucial skill for students and professionals alike. A polynomial equation is an equation in which the unknown variable (usually denoted as x) is raised to various powers, and the coefficients of these powers are constants. In this article, we will focus on solving a specific polynomial equation, $x^3 - 21x = -20$, and explore the rational root theorem to identify potential roots.

The Rational Root Theorem

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The rational root theorem is a fundamental concept in algebra that helps us identify potential roots of a polynomial equation. According to this theorem, if a rational number p/q is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, then p must be a factor of the constant term $a_0$, and q must be a factor of the leading coefficient $a_n$. In other words, the rational root theorem states that any rational root of a polynomial equation must be of the form p/q, where p is a factor of the constant term, and q is a factor of the leading coefficient.

Applying the Rational Root Theorem to the Given Equation

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Now, let's apply the rational root theorem to the given equation, $x^3 - 21x = -20$. To do this, we need to rewrite the equation in the standard form, $ax^3 + bx^2 + cx + d = 0$. By moving the constant term to the left-hand side, we get $x^3 - 21x + 20 = 0$. Now, we can identify the leading coefficient (a) as 1, and the constant term (d) as 20.

Identifying Potential Roots

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Using the rational root theorem, we can identify potential roots of the polynomial equation. The factors of the constant term (20) are ±1, ±2, ±4, ±5, ±10, and ±20. The factors of the leading coefficient (1) are only ±1. Therefore, the potential roots of the polynomial equation are ±1, ±2, ±4, ±5, ±10, and ±20.

Evaluating the Options

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Now, let's evaluate the options given in the problem. We are asked to identify which number is a potential root of the polynomial equation. The options are:

A. -7 B. 0 C. 1 D. 3

Analyzing Each Option

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Let's analyze each option to determine if it is a potential root of the polynomial equation.

Option A: -7

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We can substitute -7 into the polynomial equation to see if it satisfies the equation. Plugging in -7, we get:

$(-7)^3 - 21(-7) + 20 = -343 + 147 + 20 = -176$

Since -176 is not equal to 0, -7 is not a root of the polynomial equation.

Option B: 0

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We can substitute 0 into the polynomial equation to see if it satisfies the equation. Plugging in 0, we get:

$(0)^3 - 21(0) + 20 = 0 - 0 + 20 = 20$

Since 20 is not equal to 0, 0 is not a root of the polynomial equation.

Option C: 1

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We can substitute 1 into the polynomial equation to see if it satisfies the equation. Plugging in 1, we get:

$(1)^3 - 21(1) + 20 = 1 - 21 + 20 = 0$

Since 0 is equal to 0, 1 is a root of the polynomial equation.

Option D: 3

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We can substitute 3 into the polynomial equation to see if it satisfies the equation. Plugging in 3, we get:

$(3)^3 - 21(3) + 20 = 27 - 63 + 20 = -16$

Since -16 is not equal to 0, 3 is not a root of the polynomial equation.

Conclusion

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In conclusion, the rational root theorem helps us identify potential roots of a polynomial equation. By applying this theorem to the given equation, $x^3 - 21x = -20$, we can identify potential roots as ±1, ±2, ±4, ±5, ±10, and ±20. Evaluating the options given in the problem, we find that only option C, 1, is a potential root of the polynomial equation.

Final Answer

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The final answer is: $\boxed{1}$

===========================================================

Understanding Polynomial Equations

---

Polynomial equations are a fundamental concept in algebra, and solving them is a crucial skill for students and professionals alike. A polynomial equation is an equation in which the unknown variable (usually denoted as x) is raised to various powers, and the coefficients of these powers are constants. In this article, we will focus on solving a specific polynomial equation, $x^3 - 21x = -20$, and explore the rational root theorem to identify potential roots.

The Rational Root Theorem

---

The rational root theorem is a fundamental concept in algebra that helps us identify potential roots of a polynomial equation. According to this theorem, if a rational number p/q is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, then p must be a factor of the constant term $a_0$, and q must be a factor of the leading coefficient $a_n$. In other words, the rational root theorem states that any rational root of a polynomial equation must be of the form p/q, where p is a factor of the constant term, and q is a factor of the leading coefficient.

Applying the Rational Root Theorem to the Given Equation

---

Now, let's apply the rational root theorem to the given equation, $x^3 - 21x = -20$. To do this, we need to rewrite the equation in the standard form, $ax^3 + bx^2 + cx + d = 0$. By moving the constant term to the left-hand side, we get $x^3 - 21x + 20 = 0$. Now, we can identify the leading coefficient (a) as 1, and the constant term (d) as 20.

Identifying Potential Roots

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Using the rational root theorem, we can identify potential roots of the polynomial equation. The factors of the constant term (20) are ±1, ±2, ±4, ±5, ±10, and ±20. The factors of the leading coefficient (1) are only ±1. Therefore, the potential roots of the polynomial equation are ±1, ±2, ±4, ±5, ±10, and ±20.

Evaluating the Options

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Now, let's evaluate the options given in the problem. We are asked to identify which number is a potential root of the polynomial equation. The options are:

A. -7 B. 0 C. 1 D. 3

Analyzing Each Option

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Let's analyze each option to determine if it is a potential root of the polynomial equation.

Option A: -7

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We can substitute -7 into the polynomial equation to see if it satisfies the equation. Plugging in -7, we get:

$(-7)^3 - 21(-7) + 20 = -343 + 147 + 20 = -176$

Since -176 is not equal to 0, -7 is not a root of the polynomial equation.

Option B: 0

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We can substitute 0 into the polynomial equation to see if it satisfies the equation. Plugging in 0, we get:

$(0)^3 - 21(0) + 20 = 0 - 0 + 20 = 20$

Since 20 is not equal to 0, 0 is not a root of the polynomial equation.

Option C: 1

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We can substitute 1 into the polynomial equation to see if it satisfies the equation. Plugging in 1, we get:

$(1)^3 - 21(1) + 20 = 1 - 21 + 20 = 0$

Since 0 is equal to 0, 1 is a root of the polynomial equation.

Option D: 3

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We can substitute 3 into the polynomial equation to see if it satisfies the equation. Plugging in 3, we get:

$(3)^3 - 21(3) + 20 = 27 - 63 + 20 = -16$

Since -16 is not equal to 0, 3 is not a root of the polynomial equation.

Conclusion

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In conclusion, the rational root theorem helps us identify potential roots of a polynomial equation. By applying this theorem to the given equation, $x^3 - 21x = -20$, we can identify potential roots as ±1, ±2, ±4, ±5, ±10, and ±20. Evaluating the options given in the problem, we find that only option C, 1, is a potential root of the polynomial equation.

Final Answer

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The final answer is: $\boxed{1}$

Q&A: Solving Polynomial Equations

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Q: What is a polynomial equation?

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A: A polynomial equation is an equation in which the unknown variable (usually denoted as x) is raised to various powers, and the coefficients of these powers are constants.

Q: What is the rational root theorem?

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A: The rational root theorem is a fundamental concept in algebra that helps us identify potential roots of a polynomial equation. According to this theorem, if a rational number p/q is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, then p must be a factor of the constant term $a_0$, and q must be a factor of the leading coefficient $a_n$.

Q: How do I apply the rational root theorem to a polynomial equation?

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A: To apply the rational root theorem to a polynomial equation, you need to rewrite the equation in the standard form, $ax^3 + bx^2 + cx + d = 0$. Then, identify the leading coefficient (a) and the constant term (d). The factors of the constant term are potential roots of the polynomial equation.

Q: What are the potential roots of the polynomial equation $x^3 - 21x = -20$?

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A: The potential roots of the polynomial equation $x^3 - 21x = -20$ are ±1, ±2, ±4, ±5, ±10, and ±20.

Q: How do I evaluate the options given in the problem?

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A: To evaluate the options given in the problem, substitute each option into the polynomial equation and see if it satisfies the equation. If the result is equal to 0, then the option is a root of the polynomial equation.

Q: What is the final answer to the problem?

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A: The final answer to the problem is 1.

Q: What is the rational root theorem used for?

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A: The rational root theorem is used to identify potential roots of a polynomial equation.

Q: How do I use the rational root theorem to solve a polynomial equation?

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A: To use the rational root theorem to solve a polynomial equation, apply the theorem to the equation, identify the potential roots, and then evaluate the options given in the problem to determine the final answer.