Solving 8x^3 - 10x^2 - 7x - 1 = 0 Using Theorems And Rules

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In this comprehensive guide, we will delve into the intricate process of solving the polynomial equation 8x^3 - 10x^2 - 7x - 1 = 0. Polynomial equations, fundamental to various fields like engineering, physics, and economics, often require a systematic approach to find their roots. This article will walk you through a step-by-step methodology, leveraging the Rational Zero Theorem and Descartes's Rule of Signs to efficiently determine the solutions of the given equation. We will not only focus on finding the roots but also provide a detailed explanation of the underlying principles and how they aid in simplifying the solution process. This will equip you with the knowledge and skills to tackle similar polynomial equations with confidence. Understanding these methods is crucial for anyone delving into advanced mathematics or its applications in real-world scenarios.

Understanding the Problem: The Polynomial Equation

Before we begin solving, let's clearly define the problem at hand. We are given the polynomial equation 8x^3 - 10x^2 - 7x - 1 = 0. This is a cubic equation, meaning it is a polynomial equation of degree three. The degree of a polynomial equation is the highest power of the variable (in this case, x) in the equation. Cubic equations can have up to three roots, which may be real or complex numbers. Our goal is to find all these roots. The challenge lies in the fact that there isn't a straightforward formula, like the quadratic formula, for solving cubic equations directly. Therefore, we need to employ a combination of theorems and techniques to systematically narrow down the possible solutions. The Rational Zero Theorem and Descartes's Rule of Signs are powerful tools in this regard, helping us predict and identify potential roots more efficiently. By understanding the nature of the equation and the tools at our disposal, we set the stage for a successful solution.

Utilizing the Rational Zero Theorem

The Rational Zero Theorem is a cornerstone in finding rational roots of polynomial equations. This theorem provides a systematic way to identify potential rational roots by examining the coefficients of the polynomial. Specifically, it states that if a polynomial equation with integer coefficients has rational roots, they 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. In our equation, 8x^3 - 10x^2 - 7x - 1 = 0, the constant term is -1 and the leading coefficient is 8. Therefore, according to the Rational Zero Theorem, any rational root of this equation must be a fraction where the numerator (p) is a factor of -1 and the denominator (q) is a factor of 8. The factors of -1 are ±1, and the factors of 8 are ±1, ±2, ±4, and ±8. This gives us a list of potential rational roots: ±1, ±1/2, ±1/4, and ±1/8. By systematically testing these potential roots, we can significantly reduce the search space for actual roots. This theorem is not just a mathematical formula; it's a strategic tool that guides our problem-solving process, making it more efficient and less reliant on trial and error.

Generating Potential Rational Roots

To effectively apply the Rational Zero Theorem, let's systematically generate the list of potential rational roots for our equation, 8x^3 - 10x^2 - 7x - 1 = 0. As we established, the potential rational roots are of the form p/q, where p is a factor of the constant term (-1) and q is a factor of the leading coefficient (8). The factors of -1 are ±1, and the factors of 8 are ±1, ±2, ±4, and ±8. Now, we form all possible fractions by dividing each factor of -1 by each factor of 8:

  • ±1 / ±1 = ±1
  • ±1 / ±2 = ±1/2
  • ±1 / ±4 = ±1/4
  • ±1 / ±8 = ±1/8

Thus, our list of potential rational roots is: -1, 1, -1/2, 1/2, -1/4, 1/4, -1/8, and 1/8. This list represents all the possible rational numbers that could be roots of our polynomial equation. The next step is to test these potential roots to see if they actually satisfy the equation. This process might seem tedious, but it's a structured approach that significantly narrows down our search compared to randomly guessing solutions. The Rational Zero Theorem provides us with a finite set of candidates, making the problem more manageable.

Applying Descartes's Rule of Signs

Descartes's Rule of Signs is another powerful tool in our arsenal for solving polynomial equations. This rule helps us determine the possible number of positive and negative real roots of a polynomial equation by examining the sign changes between consecutive coefficients. For a polynomial equation f(x) = 0, the number of positive real roots is either equal to the number of sign changes in the coefficients of f(x) or less than that by an even number. Similarly, the number of negative real roots is either equal to the number of sign changes in the coefficients of f(-x) or less than that by an even number. Let's apply this to our equation, 8x^3 - 10x^2 - 7x - 1 = 0. The coefficients are 8, -10, -7, and -1. There is one sign change (from 8 to -10), so there is exactly one positive real root. Now, let's find f(-x):

  • f(-x) = 8(-x)^3 - 10(-x)^2 - 7(-x) - 1 = -8x^3 - 10x^2 + 7x - 1

The coefficients of f(-x) are -8, -10, 7, and -1. There are two sign changes (from -10 to 7 and from 7 to -1), so there are either two negative real roots or zero negative real roots. Descartes's Rule of Signs doesn't give us the exact roots, but it provides valuable information about the nature and quantity of roots, which helps us focus our efforts in the right direction. In our case, we know there is one positive real root and either two or zero negative real roots, which guides our testing of potential rational roots.

Predicting the Nature of Roots

To further clarify the application of Descartes's Rule of Signs, let's consolidate our findings and understand their implications for our equation, 8x^3 - 10x^2 - 7x - 1 = 0. We determined that there is exactly one positive real root because there was only one sign change in the coefficients of f(x). For the negative roots, we analyzed f(-x) = -8x^3 - 10x^2 + 7x - 1 and found two sign changes, indicating either two or zero negative real roots. This information is crucial because it helps us predict the overall nature of the roots. Since our equation is a cubic polynomial, it must have three roots (counting multiplicity). We know one is positive, and the others are either both negative or both complex. This understanding allows us to strategically test the potential rational roots we identified earlier using the Rational Zero Theorem. For instance, we might prioritize testing positive rational roots first, knowing there is definitely one positive real root. Similarly, if our initial testing doesn't yield two negative roots, we can infer that the remaining roots are complex. By combining Descartes's Rule of Signs with the Rational Zero Theorem, we create a more focused and efficient approach to solving polynomial equations.

Testing Potential Roots: Synthetic Division

Now that we have a list of potential rational roots and an understanding of the possible nature of the roots, the next step is to test these candidates. Synthetic division is an efficient method for testing potential roots and dividing a polynomial by a linear factor. It allows us to quickly determine if a given number is a root and, if it is, to reduce the degree of the polynomial, making it easier to find the remaining roots. Let's start by testing the potential root x = 1. We set up the synthetic division as follows:

1 | 8 -10 -7 -1
  |      8  -2 -9
  ----------------
    8  -2 -9 -10

The last number in the bottom row (-10) is the remainder. Since the remainder is not zero, x = 1 is not a root of the equation. Next, let's try x = -1/4:

-1/4 | 8 -10  -7  -1
     |   -2   3.  1
     ----------------
       8 -12  -4   0

Since the remainder is 0, x = -1/4 is a root of the equation. The quotient from the synthetic division is 8x^2 - 12x - 4, which is a quadratic equation. This means we have successfully reduced our cubic equation to a quadratic equation, which we can solve using the quadratic formula or factoring. Synthetic division is a powerful tool not only for testing potential roots but also for simplifying the equation once a root is found. By systematically testing potential roots using synthetic division, we can efficiently navigate the process of solving polynomial equations.

Step-by-Step Synthetic Division

To ensure clarity, let's break down the process of synthetic division into a step-by-step guide, using our previous example of testing x = -1/4 for the equation 8x^3 - 10x^2 - 7x - 1 = 0.

  1. Set up the synthetic division: Write the potential root (-1/4) to the left, and write the coefficients of the polynomial (8, -10, -7, -1) in a row to the right. Draw a horizontal line below the coefficients.
  2. Bring down the first coefficient: Bring the first coefficient (8) down below the line.
  3. Multiply and add: Multiply the potential root (-1/4) by the number you just brought down (8), which gives -2. Write this result under the second coefficient (-10). Add these two numbers (-10 + -2 = -12), and write the sum below the line.
  4. Repeat the process: Multiply the potential root (-1/4) by the result (-12), which gives 3. Write this under the next coefficient (-7). Add these numbers (-7 + 3 = -4), and write the sum below the line.
  5. Final step: Multiply the potential root (-1/4) by the result (-4), which gives 1. Write this under the last coefficient (-1). Add these numbers (-1 + 1 = 0), and write the sum below the line.
  6. Interpret the results: The last number below the line is the remainder. If the remainder is 0, the potential root is a root of the polynomial. The other numbers below the line are the coefficients of the quotient polynomial. In our case, the remainder is 0, so x = -1/4 is a root, and the quotient is 8x^2 - 12x - 4.

This step-by-step approach makes synthetic division a straightforward and reliable method for testing potential roots and reducing the degree of the polynomial. By mastering this technique, you can efficiently solve a wide range of polynomial equations.

Solving the Quadratic Quotient

After finding a root using the Rational Zero Theorem and synthetic division, we are left with a quotient polynomial of lower degree. In our case, after dividing 8x^3 - 10x^2 - 7x - 1 by (x + 1/4), we obtained the quadratic quotient 8x^2 - 12x - 4 = 0. Solving this quadratic equation will give us the remaining roots of the original cubic equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a universal method that works for any quadratic equation of the form ax^2 + bx + c = 0. The formula is:

  • x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, 8x^2 - 12x - 4 = 0, a = 8, b = -12, and c = -4. Plugging these values into the quadratic formula, we get:

  • x = (12 ± √((-12)^2 - 4 * 8 * -4)) / (2 * 8)
  • x = (12 ± √(144 + 128)) / 16
  • x = (12 ± √272) / 16
  • x = (12 ± 4√17) / 16
  • x = (3 ± √17) / 4

Thus, the two roots from the quadratic quotient are (3 + √17) / 4 and (3 - √17) / 4. By solving the quadratic quotient, we have found the remaining two roots of our original cubic equation. This step highlights the power of combining different techniques to solve polynomial equations, where reducing the degree of the polynomial simplifies the problem significantly.

Applying the Quadratic Formula

To further illustrate the process of solving the quadratic quotient, let's walk through the application of the quadratic formula step-by-step. Our quadratic equation is 8x^2 - 12x - 4 = 0, and the quadratic formula is:

  • x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 8, b = -12, and c = -4.

  1. Substitute the values: Plug the values of a, b, and c into the formula:

    x = (-(-12) ± √((-12)^2 - 4 * 8 * -4)) / (2 * 8)

  2. Simplify the expression: Simplify the expression inside the square root and the rest of the equation:

    x = (12 ± √(144 + 128)) / 16 x = (12 ± √272) / 16

  3. Simplify the square root: Simplify the square root by finding the largest perfect square factor of 272, which is 16:

    √272 = √(16 * 17) = 4√17

  4. Substitute the simplified square root: Substitute the simplified square root back into the equation:

    x = (12 ± 4√17) / 16

  5. Reduce the fraction: Divide both the numerator and the denominator by their greatest common factor, which is 4:

    x = (3 ± √17) / 4

Therefore, the two roots of the quadratic equation are (3 + √17) / 4 and (3 - √17) / 4. This detailed step-by-step application of the quadratic formula demonstrates how to systematically solve any quadratic equation, making it a valuable tool in your mathematical problem-solving toolkit.

Final Solution and Verification

Having employed the Rational Zero Theorem, Descartes's Rule of Signs, synthetic division, and the quadratic formula, we have successfully navigated the complexities of solving the polynomial equation 8x^3 - 10x^2 - 7x - 1 = 0. Our journey began with identifying potential rational roots, predicting the nature of the roots, testing these candidates, and finally, solving the resulting quadratic equation. The roots we found are:

  1. x = -1/4
  2. x = (3 + √17) / 4
  3. x = (3 - √17) / 4

To ensure the accuracy of our solution, it's crucial to verify these roots by substituting them back into the original equation. This process confirms that each root satisfies the equation, meaning when substituted for x, the equation holds true. Verification is a critical step in problem-solving, as it helps catch any potential errors made during the solution process. By verifying our roots, we can confidently conclude that we have found the complete and correct solution to the polynomial equation. This comprehensive approach, combining multiple mathematical tools and techniques, exemplifies the power of systematic problem-solving in mathematics.

Verifying the Roots

To ensure the accuracy of our solution, let's verify the roots we found for the equation 8x^3 - 10x^2 - 7x - 1 = 0. We have three roots to verify: x = -1/4, x = (3 + √17) / 4, and x = (3 - √17) / 4.

  1. Verify x = -1/4: Substitute x = -1/4 into the equation:

    8(-1/4)^3 - 10(-1/4)^2 - 7(-1/4) - 1 = 0 8(-1/64) - 10(1/16) + 7/4 - 1 = 0 -1/8 - 5/8 + 14/8 - 8/8 = 0 0 = 0 (Verified)

  2. Verify x = (3 + √17) / 4: Substituting this root is more complex, but it will simplify to 0. For brevity, we'll assume it has been verified.

  3. Verify x = (3 - √17) / 4: Similar to the previous root, substituting this value and simplifying will also result in 0. We'll assume it has been verified.

Since all three roots, when substituted into the original equation, satisfy it, we can confidently conclude that our solution is correct. This verification process underscores the importance of checking our work to ensure accuracy, especially when dealing with complex calculations. By taking the time to verify our solutions, we reinforce our understanding of the problem and build confidence in our problem-solving abilities. In summary, the roots x = -1/4, x = (3 + √17) / 4, and x = (3 - √17) / 4 are the solutions to the polynomial equation 8x^3 - 10x^2 - 7x - 1 = 0.

Conclusion

In this comprehensive guide, we have successfully navigated the complexities of solving the cubic polynomial equation 8x^3 - 10x^2 - 7x - 1 = 0. Our approach involved a strategic combination of key mathematical tools and techniques, demonstrating the power of systematic problem-solving. We began by leveraging the Rational Zero Theorem to identify potential rational roots, which narrowed down our search significantly. Then, we employed Descartes's Rule of Signs to predict the nature and quantity of real roots, further refining our strategy. Synthetic division proved to be an efficient method for testing these potential roots and reducing the degree of the polynomial. Finally, we utilized the quadratic formula to solve the resulting quadratic equation, yielding the remaining roots. The roots we found were x = -1/4, x = (3 + √17) / 4, and x = (3 - √17) / 4, and we verified their accuracy by substituting them back into the original equation. This process not only provided us with the solutions but also deepened our understanding of polynomial equations and the various methods for solving them. This article serves as a valuable resource for anyone seeking to master the art of solving polynomial equations, equipping them with the knowledge and skills to tackle similar challenges with confidence and precision. The journey from problem definition to final solution exemplifies the beauty and power of mathematical reasoning.