Finding All Roots Of F(x) = X^3 - 9x^2 + 26x - 24 Using The Remainder Theorem

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Introduction: Unveiling the Roots of a Cubic Polynomial

In the realm of algebra, finding the roots of a polynomial function is a fundamental task. Roots, also known as zeros, are the values of x for which the function f(x) equals zero. These roots provide crucial information about the behavior and characteristics of the polynomial. In this article, we delve into the process of determining all the roots of the cubic polynomial function f(x) = x^3 - 9x^2 + 26x - 24, given that one of its roots is x = 2. We will employ the Remainder Theorem and the technique of polynomial division to systematically uncover the remaining roots. Understanding these methods is essential for anyone seeking to master polynomial algebra and its applications.

This exploration into polynomial roots is not merely an academic exercise; it has practical implications across various fields. From engineering and physics to economics and computer science, polynomial functions are used to model real-world phenomena. Identifying the roots of these functions allows us to solve equations, optimize processes, and make informed predictions. By mastering the techniques discussed in this article, readers will gain valuable tools for tackling a wide range of problems.

To begin, we will leverage the Remainder Theorem, a powerful tool in polynomial algebra, to confirm that x = 2 is indeed a root of the given function. This theorem provides a shortcut for evaluating polynomials at specific values and determining if those values are roots. Next, we will perform polynomial division, a process that allows us to factor out the known root and reduce the cubic polynomial to a quadratic polynomial. This reduction is a crucial step, as it simplifies the problem and allows us to apply familiar techniques for finding the roots of quadratic equations. Finally, we will solve the resulting quadratic equation, revealing the remaining two roots of the original cubic polynomial. By the end of this article, readers will have a clear understanding of how to systematically find all the roots of a polynomial function when one root is known, equipping them with a valuable skill for mathematical problem-solving.

Utilizing the Remainder Theorem: Confirming the Root

The Remainder Theorem provides a powerful shortcut for determining if a given value is a root of a polynomial function. It states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). In simpler terms, to check if a value a is a root of f(x), we can substitute a into the function and see if the result is zero. If f(a) = 0, then a is a root of the polynomial.

In our case, we are given that x = 2 is a root of the polynomial f(x) = x^3 - 9x^2 + 26x - 24. To confirm this using the Remainder Theorem, we substitute x = 2 into the function:

f(2) = (2)^3 - 9(2)^2 + 26(2) - 24 f(2) = 8 - 9(4) + 52 - 24 f(2) = 8 - 36 + 52 - 24 f(2) = 0

As we can see, f(2) = 0, which confirms that x = 2 is indeed a root of the polynomial f(x). This result is significant because it tells us that (x - 2) is a factor of the polynomial. This understanding is crucial for the next step in our process, which involves using polynomial division to factor out the known root and simplify the problem.

The Remainder Theorem not only helps us confirm known roots but also provides a valuable tool for exploring potential roots. If we were not given a root to begin with, we could use the Remainder Theorem in conjunction with the Rational Root Theorem to test potential rational roots. The Rational Root Theorem provides a list of possible rational roots based on the coefficients of the polynomial. By testing these potential roots using the Remainder Theorem, we can efficiently identify actual roots and begin the process of factoring the polynomial. This combination of theorems is a cornerstone of polynomial algebra, allowing us to systematically analyze and solve polynomial equations. The ability to quickly verify roots using the Remainder Theorem is a fundamental skill for anyone working with polynomials, as it streamlines the process of finding all the solutions to a polynomial equation.

Polynomial Division: Reducing the Cubic to a Quadratic

Since we have confirmed that x = 2 is a root of f(x) = x^3 - 9x^2 + 26x - 24, we know that (x - 2) is a factor of the polynomial. To find the other factor, we can use polynomial long division. This process is analogous to long division with numbers and allows us to divide a polynomial by another polynomial of equal or lower degree.

We set up the long division as follows:

 x^2 - 7x + 12
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24

Now, we perform the division step by step:

  1. Divide the first term of the dividend (x^3) by the first term of the divisor (x), which gives us x^2. Write x^2 above the division line.
  2. Multiply the divisor (x - 2) by x^2, which gives us x^3 - 2x^2. Write this below the dividend and subtract.
 x^2
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24
       x^3 - 2x^2
       ----------
          -7x^2
  1. Bring down the next term from the dividend (+26x).
 x^2
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24
       x^3 - 2x^2
       ----------
          -7x^2 + 26x
  1. Divide the first term of the new dividend (-7x^2) by the first term of the divisor (x), which gives us -7x. Write -7x above the division line.
  2. Multiply the divisor (x - 2) by -7x, which gives us -7x^2 + 14x. Write this below the new dividend and subtract.
 x^2 - 7x
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24
       x^3 - 2x^2
       ----------
          -7x^2 + 26x
          -7x^2 + 14x
          ----------
                  12x
  1. Bring down the last term from the dividend (-24).
 x^2 - 7x
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24
       x^3 - 2x^2
       ----------
          -7x^2 + 26x
          -7x^2 + 14x
          ----------
                  12x - 24
  1. Divide the first term of the new dividend (12x) by the first term of the divisor (x), which gives us 12. Write +12 above the division line.
  2. Multiply the divisor (x - 2) by 12, which gives us 12x - 24. Write this below the new dividend and subtract.
 x^2 - 7x + 12
------------------------
x - 2 | x^3 - 9x^2 + 26x - 24
       x^3 - 2x^2
       ----------
          -7x^2 + 26x
          -7x^2 + 14x
          ----------
                  12x - 24
                  12x - 24
                  ----------
                          0

The remainder is 0, which confirms that (x - 2) is a factor. The quotient is x^2 - 7x + 12. This means we can now write the original polynomial as:

f(x) = (x - 2)(x^2 - 7x + 12)

By performing polynomial division, we have successfully reduced the cubic polynomial to a product of a linear factor (x - 2) and a quadratic factor (x^2 - 7x + 12). This reduction is a crucial step because finding the roots of a quadratic equation is a much simpler task than finding the roots of a cubic equation. We can now use various methods, such as factoring, completing the square, or the quadratic formula, to find the roots of the quadratic factor. This process of reducing the degree of the polynomial through division is a fundamental technique in algebra, allowing us to solve complex polynomial equations by breaking them down into simpler components. The ability to perform polynomial division efficiently and accurately is an essential skill for anyone working with polynomial functions.

Solving the Quadratic: Finding the Remaining Roots

Having reduced our cubic polynomial to f(x) = (x - 2)(x^2 - 7x + 12), we now need to find the roots of the quadratic factor x^2 - 7x + 12. There are several methods we can use to solve quadratic equations, including factoring, completing the square, and the quadratic formula. In this case, factoring is the most straightforward approach.

We are looking for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. Therefore, we can factor the quadratic as follows:

x^2 - 7x + 12 = (x - 3)(x - 4)

Now we have the complete factorization of the original polynomial:

f(x) = (x - 2)(x - 3)(x - 4)

To find the roots, we set each factor equal to zero and solve for x:

x - 2 = 0 => x = 2 x - 3 = 0 => x = 3 x - 4 = 0 => x = 4

Thus, the roots of the polynomial f(x) = x^3 - 9x^2 + 26x - 24 are x = 2, x = 3, and x = 4. These are the values of x for which the function f(x) equals zero. They represent the points where the graph of the polynomial intersects the x-axis.

The process of factoring the quadratic equation is a powerful technique that relies on recognizing patterns and relationships between the coefficients of the quadratic and its factors. However, not all quadratic equations can be easily factored. In such cases, the quadratic formula provides a reliable method for finding the roots. The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Applying this formula to x^2 - 7x + 12 = 0, we would obtain the same roots, x = 3 and x = 4. Understanding and being able to apply different methods for solving quadratic equations is crucial for mastering polynomial algebra. The ability to choose the most efficient method for a given problem, whether it's factoring, completing the square, or using the quadratic formula, is a hallmark of a proficient problem solver. By finding the roots of the quadratic factor, we have successfully completed the process of finding all the roots of the original cubic polynomial, demonstrating the power of combining the Remainder Theorem, polynomial division, and quadratic equation solving techniques.

Conclusion: The Complete Set of Roots

In this article, we embarked on a journey to find all the roots of the cubic polynomial function f(x) = x^3 - 9x^2 + 26x - 24, given that one of its roots is x = 2. We successfully navigated this problem by employing a combination of powerful techniques from polynomial algebra.

First, we utilized the Remainder Theorem to confirm that x = 2 was indeed a root of the polynomial. This theorem provided a quick and efficient way to verify the given information and establish a foundation for our subsequent steps. The Remainder Theorem's ability to link the value of a polynomial at a specific point to the remainder upon division is a testament to its utility in polynomial analysis.

Next, we performed polynomial long division to divide the original cubic polynomial by (x - 2). This process reduced the cubic polynomial to a product of a linear factor (x - 2) and a quadratic factor x^2 - 7x + 12. This reduction was a crucial step, as it simplified the problem and allowed us to focus on finding the roots of the quadratic factor. Polynomial division is a fundamental skill in algebra, enabling us to factor polynomials and solve equations of higher degrees.

Finally, we solved the quadratic equation x^2 - 7x + 12 = 0 by factoring. This revealed the remaining two roots of the original polynomial, x = 3 and x = 4. By setting each factor to zero, we systematically uncovered all the values of x that make the polynomial equal to zero.

Therefore, the complete set of roots for the polynomial function f(x) = x^3 - 9x^2 + 26x - 24 is x = 2, x = 3, and x = 4. These roots provide a comprehensive understanding of the polynomial's behavior and its relationship to the x-axis. They represent the points where the graph of the polynomial intersects the x-axis, and they are essential for solving equations, modeling real-world phenomena, and performing further analysis of the polynomial function.

This process of finding all the roots of a polynomial when one root is known highlights the interconnectedness of various algebraic concepts and techniques. The Remainder Theorem, polynomial division, and quadratic equation solving methods work in concert to provide a systematic approach to solving polynomial equations. Mastering these techniques is essential for anyone seeking to excel in algebra and its applications. The ability to analyze polynomials, identify their roots, and understand their behavior is a valuable skill that extends far beyond the classroom, finding applications in various fields of science, engineering, and mathematics.

Keywords

Polynomial roots, Remainder Theorem, Polynomial division, Quadratic equation, Factoring, Cubic polynomial, Roots of a function, Zeros of a function, Algebraic techniques, Solving polynomial equations