Finding All Roots Of F(x) = X³ + 10x² - 25x - 250 Using The Remainder Theorem

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One root of the cubic function f(x) = x³ + 10x² - 25x - 250 is given as x = -10. In this comprehensive article, we will delve into the process of finding all the roots of this function using the Remainder Theorem and polynomial division. Understanding how to solve such problems is crucial in algebra and calculus, as it lays the groundwork for more advanced mathematical concepts. This article will walk you through the steps, explain the underlying principles, and provide a clear, step-by-step solution. Whether you're a student grappling with polynomial equations or just someone keen to understand mathematical problem-solving, this guide is tailored to make the process clear and accessible.

Understanding the Problem

The given function is a cubic polynomial, which means it has a degree of 3. By the Fundamental Theorem of Algebra, a cubic polynomial has exactly three roots, which may be real or complex. We are already provided with one root, x = -10. Our mission is to find the remaining two roots. To achieve this, we will leverage the Remainder Theorem and polynomial division. This approach not only helps in solving the problem but also reinforces key concepts in polynomial algebra.

The Remainder Theorem and Factor Theorem

Before diving into the solution, it's essential to understand the Remainder Theorem and its close relative, the Factor Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - c, then the remainder is f(c). The Factor Theorem is a special case of the Remainder Theorem, which states that if f(c) = 0, then x - c is a factor of f(x). In our case, since x = -10 is a root of f(x) = x³ + 10x² - 25x - 250, we know that f(-10) = 0. This implies that (x + 10) is a factor of f(x). This understanding is a cornerstone for solving polynomial equations and simplifies the process of finding roots significantly.

Applying Polynomial Division

Now that we know (x + 10) is a factor, we can perform polynomial division to find the other factor. Dividing f(x) = x³ + 10x² - 25x - 250 by (x + 10) will give us a quadratic polynomial, which will be much easier to solve for the remaining roots. Polynomial division is a method of dividing a polynomial by another polynomial of a lower or equal degree. It's a systematic way to break down a higher-degree polynomial into simpler factors, which is invaluable for finding roots and understanding polynomial behavior. The steps involved in polynomial division are similar to long division with numbers, but with algebraic expressions. This process helps reduce the complexity of the problem and sets the stage for finding the remaining roots.

Here's how we perform the polynomial division:

  1. Set up the long division with x³ + 10x² - 25x - 250 as the dividend and (x + 10) as the divisor.
  2. Divide the first term of the dividend (x³) by the first term of the divisor (x), which gives x². This is the first term of the quotient.
  3. Multiply the divisor (x + 10) by x² to get x³ + 10x².
  4. Subtract this result from the dividend: (x³ + 10x² - 25x - 250) - (x³ + 10x²) = -25x - 250.
  5. Bring down the next term, which is -250, resulting in -25x - 250.
  6. Divide the first term of the new dividend (-25x) by the first term of the divisor (x), which gives -25. This is the next term of the quotient.
  7. Multiply the divisor (x + 10) by -25 to get -25x - 250.
  8. Subtract this result from the new dividend: (-25x - 250) - (-25x - 250) = 0.

The remainder is 0, as expected, and the quotient is x² - 25. This means that f(x) can be factored as (x + 10)(x² - 25).

Solving the Quadratic Equation

Now we have f(x) = (x + 10)(x² - 25). To find the remaining roots, we need to solve the quadratic equation x² - 25 = 0. This is a difference of squares, which can be factored as (x - 5)(x + 5). Therefore, the equation becomes:

(x + 10)(x - 5)(x + 5) = 0

The roots are the values of x that make each factor equal to zero. So, we have:

  • x + 10 = 0 => x = -10
  • x - 5 = 0 => x = 5
  • x + 5 = 0 => x = -5

Thus, the roots of the function f(x) = x³ + 10x² - 25x - 250 are x = -10, x = 5, and x = -5.

Final Answer

The roots of the function f(x) = x³ + 10x² - 25x - 250 are -10, 5, and -5. This solution was obtained by using the Remainder Theorem to identify a factor, polynomial division to simplify the cubic function into a quadratic function, and then solving the quadratic equation to find the remaining roots. Understanding and applying these techniques are essential for solving polynomial equations and mastering algebra.

Conclusion

In summary, finding the roots of a polynomial function like f(x) = x³ + 10x² - 25x - 250 involves several key steps: utilizing the Remainder Theorem to identify factors, applying polynomial division to reduce the complexity of the equation, and solving the resulting lower-degree polynomial. In this case, by starting with the given root x = -10, we successfully factored the cubic polynomial and found the remaining roots, x = 5 and x = -5. This process highlights the interconnectedness of algebraic concepts and the power of systematic problem-solving in mathematics. The ability to find roots of polynomial functions is not just a theoretical exercise; it has practical applications in various fields, including engineering, physics, and computer science. Therefore, mastering these techniques is a valuable skill for anyone pursuing studies or a career in these areas. Furthermore, this example showcases the beauty of mathematics, where seemingly complex problems can be broken down into manageable steps with the right tools and understanding. The combination of the Remainder Theorem, polynomial division, and factoring techniques provides a robust method for tackling polynomial equations, and this knowledge empowers us to approach similar problems with confidence and clarity.