Finding Roots Of Cubic Function F(x) = X³ - 9x² + 26x - 24 Using The Remainder Theorem
One of the fundamental problems in algebra is finding the roots (or zeros) of a polynomial function. A root of a function f(x) is a value of x that makes f(x) equal to zero. In this article, we will explore a method to find all the roots of the cubic function f(x) = x³ - 9x² + 26x - 24, given that one of the roots is x = 2. We will use the Remainder Theorem and polynomial division as our primary tools. This approach is a cornerstone of polynomial algebra, allowing us to break down complex equations into more manageable forms and identify all solutions. Mastering these techniques is essential for anyone delving into higher-level mathematics, engineering, or any field that relies on mathematical modeling.
Understanding the Remainder Theorem
The Remainder Theorem is a crucial concept in polynomial algebra. It states that if a polynomial f(x) is divided by x - a, the remainder is f(a). In simpler terms, if we substitute x = a into the polynomial f(x), the result we get is the same as the remainder when f(x) is divided by x - a. This theorem provides a direct link between the value of a polynomial at a specific point and the remainder of a polynomial division. This connection allows us to efficiently check if a given value is a root of the polynomial. If f(a) = 0, then x - a is a factor of f(x), and a is a root. This principle is the foundation for factoring polynomials and finding their roots, making the Remainder Theorem a powerful tool in algebraic problem-solving. It’s not just a theoretical concept; it has practical applications in various fields, including computer science, where polynomial manipulation is essential for algorithm design and optimization.
In our case, we're given that x = 2 is a root of the function f(x) = x³ - 9x² + 26x - 24. This means that f(2) = 0. According to the Remainder Theorem, if we divide f(x) by (x - 2), the remainder should be zero. This confirms that (x - 2) is a factor of f(x). To find the other roots, we first need to perform polynomial division. Polynomial division is an extension of the familiar long division method, but applied to polynomials. It allows us to divide one polynomial by another, resulting in a quotient and a remainder. In this context, polynomial division is crucial for reducing the cubic polynomial into a quadratic, which is easier to solve. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, subtracting from the dividend, and repeating the process until the degree of the remainder is less than the degree of the divisor. This step-by-step method is essential for simplifying polynomials and revealing their underlying factors, ultimately aiding in the determination of roots.
Polynomial Division
Let's divide f(x) = x³ - 9x² + 26x - 24 by (x - 2). This process will help us factor the cubic polynomial into a product of a linear and a quadratic term. Polynomial division is similar to long division with numbers, but we're dealing with algebraic expressions. We set up the division problem like this:
x² - 7x + 12
x - 2 | x³ - 9x² + 26x - 24
- (x³ - 2x²)
------------------
-7x² + 26x
- (-7x² + 14x)
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12x - 24
- (12x - 24)
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0
The quotient we obtain is x² - 7x + 12. The remainder is 0, which confirms the Remainder Theorem since we knew x = 2 was a root. Now we can express f(x) as:
f(x) = (x - 2)(x² - 7x + 12)
This factorization is a critical step in finding all the roots of the function. By reducing the cubic polynomial to a product of a linear term (x - 2) and a quadratic term (x² - 7x + 12), we simplify the problem of finding roots. The roots of the linear term are straightforward to determine, and the quadratic term can be further factored or solved using the quadratic formula. This process of breaking down a higher-degree polynomial into lower-degree factors is a fundamental technique in algebra, allowing us to systematically find all possible solutions. The ability to perform polynomial division and factorization is essential for solving a wide range of mathematical problems, particularly those involving polynomial equations and functions.
Finding the Remaining Roots
Now, we need to find the roots of the quadratic equation x² - 7x + 12 = 0. To find the remaining roots, we can factor the quadratic expression. Factoring a quadratic expression involves finding two binomials that, when multiplied together, give the original quadratic. This technique is based on the distributive property of multiplication and involves identifying two numbers that add up to the coefficient of the linear term and multiply to the constant term. Factoring is an efficient method for solving quadratic equations when the roots are rational numbers. It provides a direct and intuitive way to find the solutions without resorting to more complex formulas or methods. However, it's important to note that not all quadratic expressions can be easily factored, and in such cases, other methods like the quadratic formula are necessary.
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:
x² - 7x + 12 = (x - 3)(x - 4)
Setting each factor equal to zero gives us the roots:
x - 3 = 0 => x = 3 x - 4 = 0 => x = 4
Thus, the roots of the quadratic equation are x = 3 and x = 4. Combining these with the given root x = 2, we have found all three roots of the cubic function. This process highlights the power of factorization in solving polynomial equations. By breaking down the quadratic into two linear factors, we were able to quickly identify the solutions. This approach is not only efficient but also provides a clear understanding of the relationship between the roots and the factors of the polynomial. The ability to factor quadratic expressions is a fundamental skill in algebra, with wide applications in various mathematical and scientific fields.
Conclusion
The roots of the function f(x) = x³ - 9x² + 26x - 24 are x = 2, x = 3, and x = 4. We found these roots by using the Remainder Theorem to verify that x = 2 is a root, performing polynomial division to reduce the cubic to a quadratic, and then factoring the quadratic to find the remaining roots. In conclusion, finding the roots of a polynomial function is a fundamental problem in algebra, and the Remainder Theorem, combined with polynomial division and factorization, provides a powerful toolkit for solving such problems. These techniques allow us to systematically break down complex polynomials into simpler factors, revealing their roots. Understanding these methods is crucial for success in higher-level mathematics and its applications. The process we followed in this article demonstrates a general strategy for finding the roots of polynomials, which can be applied to a wide range of problems. By mastering these skills, students and practitioners can confidently tackle polynomial equations and gain a deeper understanding of algebraic concepts.
The correct answer is:
A. x = 2, x = 3, or x = 4
