Finding the Possible Rational Solutions

Oct 13, 2025 by windailys.com 100 views

Introduction

The Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational solutions of a polynomial equation. It states that if a rational number p/q is a root of the polynomial equation a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + 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 this article, we will use the Rational Zero Theorem to find the real solution(s) of the cubic equation x^3 - 3x^2 - 11x + 21 = 0.

The Rational Zero Theorem

The Rational Zero Theorem is a powerful tool for finding the possible rational solutions of a polynomial equation. It is based on the following theorem:

Theorem: If a rational number p/q is a root of the polynomial equation a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + 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.

Proof: Let p/q be a root of the polynomial equation. Then, we can write:

a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0

Multiplying both sides by q^n, we get:

a_n q^n x^n + a_(n-1) q^n x^(n-1) + ... + a_1 q^n x + a_0 q^n = 0

Since p/q is a root, we know that p/q is equal to x. Therefore, we can substitute x = p/q into the equation:

a_n q^n (p/q)^n + a_(n-1) q^n (p/q)^(n-1) + ... + a_1 q^n (p/q) + a_0 q^n = 0

Simplifying the equation, we get:

a_n p^n + a_(n-1) p^(n-1) q + ... + a_1 p q^(n-1) + a_0 q^n = 0

Since p/q is a root, we know that the left-hand side of the equation is equal to zero. Therefore, we can conclude that p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.

Finding the Possible Rational Solutions

To find the possible rational solutions of the cubic equation x^3 - 3x^2 - 11x + 21 = 0, we need to find the factors of the constant term 21 and the leading coefficient 1.

Factors of 21: The factors of 21 are ±1, ±3, ±7, and ±21.

Factors of 1: The factors of 1 are ±1.

Using the Rational Zero Theorem to Find the Real Solution(s)

Now that we have found the possible rational solutions, we can use the Rational Zero Theorem to find the real solution(s) of the cubic equation.

Step 1: Check if x = 1 is a solution.

Substituting x = 1 into the equation, we get:

(1)^3 - 3(1)^2 - 11(1) + 21 = 1 - 3 - 11 + 21 = 8 ≠ 0

Therefore, x = 1 is not a solution.

Step 2: Check if x = -1 is a solution.

Substituting x = -1 into the equation, we get:

(-1)^3 - 3(-1)^2 - 11(-1) + 21 = -1 - 3 + 11 + 21 = 28 ≠ 0

Therefore, x = -1 is not a solution.

Step 3: Check if x = 3 is a solution.

Substituting x = 3 into the equation, we get:

(3)^3 - 3(3)^2 - 11(3) + 21 = 27 - 27 - 33 + 21 = -12 ≠ 0

Therefore, x = 3 is not a solution.

Step 4: Check if x = -3 is a solution.

Substituting x = -3 into the equation, we get:

(-3)^3 - 3(-3)^2 - 11(-3) + 21 = -27 - 27 + 33 + 21 = 0

Therefore, x = -3 is a solution.

Conclusion

In this article, we used the Rational Zero Theorem to find the real solution(s) of the cubic equation x^3 - 3x^2 - 11x + 21 = 0. We found that x = -3 is a solution. The Rational Zero Theorem is a powerful tool for finding the possible rational solutions of a polynomial equation, and it can be used to solve a wide range of problems in algebra.

Final Answer

The final answer is $\boxed{-3}$ .

Introduction

In our previous article, we used the Rational Zero Theorem to find the real solution(s) of the cubic equation x^3 - 3x^2 - 11x + 21 = 0. We found that x = -3 is a solution. In this article, we will answer some frequently asked questions about the Rational Zero Theorem and its application to solving polynomial equations.

Q&A

Q: What is the Rational Zero Theorem?

A: The Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational solutions of a polynomial equation. It states that if a rational number p/q is a root of the polynomial equation a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + 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 Zero Theorem to find the possible rational solutions of a polynomial equation?

A: To apply the Rational Zero Theorem, you need to find the factors of the constant term a_0 and the leading coefficient a_n. Then, you can use these factors to find the possible rational solutions of the polynomial equation.

Q: What are the steps to follow when using the Rational Zero Theorem to find the real solution(s) of a polynomial equation?

A: The steps to follow are:

Find the factors of the constant term a_0 and the leading coefficient a_n.
Use these factors to find the possible rational solutions of the polynomial equation.
Check if each possible rational solution is actually a solution by substituting it into the equation.
If a possible rational solution is a solution, then it is a real solution of the polynomial equation.

Q: Can the Rational Zero Theorem be used to find the real solution(s) of a polynomial equation with complex coefficients?

A: No, the Rational Zero Theorem can only be used to find the real solution(s) of a polynomial equation with real coefficients.

Q: Can the Rational Zero Theorem be used to find the real solution(s) of a polynomial equation with a degree greater than 3?

A: Yes, the Rational Zero Theorem can be used to find the real solution(s) of a polynomial equation with a degree greater than 3.

Q: How do I know if a possible rational solution is actually a solution of the polynomial equation?

A: You can check if a possible rational solution is actually a solution by substituting it into the equation. If the equation is true, then the possible rational solution is a solution.

Q: Can the Rational Zero Theorem be used to find the real solution(s) of a polynomial equation with a non-integer coefficient?

A: No, the Rational Zero Theorem can only be used to find the real solution(s) of a polynomial equation with integer coefficients.

Conclusion

In this article, we answered some frequently asked questions about the Rational Zero Theorem and its application to solving polynomial equations. We hope that this article has been helpful in understanding the Rational Zero Theorem and how to use it to find the real solution(s) of a polynomial equation.

Final Answer

The final answer is $\boxed{-3}$ .