Finding Roots Of F(x) = X^3 + 10x^2 - 25x - 250 Using The Remainder Theorem

by ADMIN 76 views
Iklan Headers

Introduction

In this article, we will delve into the process of finding all the roots of the cubic function f(x) = x^3 + 10x^2 - 25x - 250. We are given that one root of the function is x = -10. To find the remaining roots, we will employ the Remainder Theorem and polynomial division. This method allows us to break down the cubic function into a product of linear factors, ultimately revealing all the values of x for which f(x) equals zero. Understanding how to find roots of polynomial functions is a fundamental concept in algebra and has wide applications in various fields, including engineering, physics, and computer science.

Understanding the Remainder Theorem

The Remainder Theorem is a crucial concept in polynomial algebra that connects the value of a polynomial at a specific point to the remainder obtained when the polynomial is divided by a linear factor. Specifically, the Remainder Theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is equal to f(c). In simpler terms, if you substitute 'c' into the polynomial, the result will be the same as the remainder you get when you perform polynomial long division by (x - c). This theorem provides a powerful shortcut for evaluating polynomials and determining factors.

For example, consider a polynomial f(x) and a number 'c'. If f(c) = 0, then the remainder when f(x) is divided by (x - c) is zero. This implies that (x - c) is a factor of f(x). Conversely, if (x - c) is a factor of f(x), then f(c) must be zero. This relationship between roots and factors is the cornerstone of our approach to finding all the roots of the given cubic function. The Remainder Theorem allows us to quickly verify if a given value is a root of the polynomial, and if it is, it helps us factorize the polynomial, making it easier to find the other roots. This theorem is not just a theoretical concept; it is a practical tool that simplifies the process of solving polynomial equations.

Applying the Remainder Theorem to Our Cubic Function

Given the cubic function f(x) = x^3 + 10x^2 - 25x - 250, we know that one root is x = -10. This means that f(-10) = 0. According to the Remainder Theorem, if x = -10 is a root, then (x + 10) must be a factor of f(x). This is because substituting x = -10 into (x + 10) gives us (-10 + 10) = 0, which is consistent with the definition of a root. To find the other factors and, consequently, the other roots, we will perform polynomial division. We will divide f(x) by (x + 10) to obtain a quadratic expression, which will be easier to solve for its roots.

Polynomial division is a systematic way of dividing one polynomial by another. In this case, we are dividing the cubic polynomial f(x) by the linear factor (x + 10). The process involves dividing the highest degree term of the dividend (x^3) by the highest degree term of the divisor (x), which gives us x^2. We then multiply the entire divisor (x + 10) by x^2 and subtract the result from the dividend. This process is repeated with the resulting polynomial until we obtain a remainder. If the remainder is zero, it confirms that (x + 10) is indeed a factor of f(x). The quotient obtained from this division will be a quadratic polynomial, which we can then further analyze to find the remaining roots. This step is crucial because it reduces the cubic equation to a quadratic equation, which can be solved using various methods, such as factoring, completing the square, or the quadratic formula.

Polynomial Division: Dividing f(x) by (x + 10)

Now, let's perform the polynomial division of f(x) = x^3 + 10x^2 - 25x - 250 by (x + 10). This process will help us find the other factor of the cubic function and eventually lead us to the remaining roots. Polynomial division is a step-by-step method, and it's important to follow each step carefully to avoid errors.

  1. Set up the division: Write the dividend (x^3 + 10x^2 - 25x - 250) inside the division symbol and the divisor (x + 10) outside.
  2. Divide the leading terms: Divide the first term of the dividend (x^3) by the first term of the divisor (x). This gives us x^2. Write x^2 above the division symbol, aligning it with the x^2 term in the dividend.
  3. Multiply and subtract: Multiply the divisor (x + 10) by x^2, which gives us x^3 + 10x^2. Subtract this result from the corresponding terms in the dividend. The x^3 and 10x^2 terms will cancel out, leaving us with -25x - 250.
  4. Bring down the next term: Bring down the next term from the dividend (-25x) to the remainder, forming the new dividend -25x - 250.
  5. Repeat the process: Divide the first term of the new dividend (-25x) by the first term of the divisor (x). This gives us -25. Write -25 above the division symbol, aligning it with the constant term in the dividend.
  6. Multiply and subtract again: Multiply the divisor (x + 10) by -25, which gives us -25x - 250. Subtract this result from the new dividend (-25x - 250). This subtraction results in a remainder of 0.

The result of the polynomial division is the quotient x^2 - 25. Since the remainder is 0, this confirms that (x + 10) is indeed a factor of f(x), and the other factor is x^2 - 25. This step is crucial because it transforms the cubic function into a product of a linear factor (x + 10) and a quadratic factor (x^2 - 25), making it easier to find the remaining roots.

Factoring the Quadratic Expression

After performing the polynomial division, we found that f(x) = (x + 10)(x^2 - 25). To find the remaining roots of the function, we need to factor the quadratic expression (x^2 - 25). This quadratic expression is a difference of squares, which is a special form that can be easily factored. Recognizing and applying factoring patterns like the difference of squares is a fundamental skill in algebra and can significantly simplify the process of solving equations.

The difference of squares pattern states that a^2 - b^2 can be factored as (a - b)(a + b). In our case, x^2 - 25 can be seen as x^2 - 5^2. Applying the difference of squares pattern, we can factor x^2 - 25 as (x - 5)(x + 5). This factorization is a critical step in finding the remaining roots because it breaks down the quadratic expression into two linear factors, each of which can be easily solved for x.

Therefore, the complete factorization of the cubic function f(x) is f(x) = (x + 10)(x - 5)(x + 5). This factorization reveals all the linear factors of the polynomial, and each factor corresponds to a root of the function. By setting each factor equal to zero, we can find the values of x that make the function equal to zero, which are the roots of the function. Understanding how to factor quadratic expressions, especially the difference of squares, is essential for solving polynomial equations and is a key concept in algebra.

Finding All the Roots

Now that we have completely factored the cubic function as f(x) = (x + 10)(x - 5)(x + 5), we can easily find all the roots. The roots of a function are the values of x for which f(x) = 0. To find these values, we set each factor of the factored form equal to zero and solve for x. This method is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero.

  1. Set each factor to zero:
    • x + 10 = 0
    • x - 5 = 0
    • x + 5 = 0
  2. Solve for x:
    • From x + 10 = 0, we subtract 10 from both sides to get x = -10.
    • From x - 5 = 0, we add 5 to both sides to get x = 5.
    • From x + 5 = 0, we subtract 5 from both sides to get x = -5.

Therefore, the roots of the function f(x) = x^3 + 10x^2 - 25x - 250 are x = -10, x = 5, and x = -5. These are the three values of x that make the function equal to zero. A cubic function, in general, can have up to three roots, and in this case, we have found all three. These roots are the points where the graph of the function intersects the x-axis. Finding the roots of a polynomial is a fundamental problem in algebra, and the ability to factor polynomials and apply the zero-product property is a crucial skill for solving such problems.

Conclusion

In summary, we have successfully found all the roots of the cubic function f(x) = x^3 + 10x^2 - 25x - 250 using the Remainder Theorem and polynomial division. We started with the given root x = -10 and used the Remainder Theorem to confirm that (x + 10) is a factor of the function. We then performed polynomial division to divide f(x) by (x + 10), which resulted in the quadratic expression x^2 - 25. We factored this quadratic expression using the difference of squares pattern, obtaining (x - 5)(x + 5). Finally, we set each factor equal to zero and solved for x to find all the roots: x = -10, x = 5, and x = -5.

This process demonstrates a powerful method for finding the roots of polynomial functions. The combination of the Remainder Theorem, polynomial division, and factoring techniques allows us to break down complex polynomials into simpler factors, making it easier to find their roots. Understanding these concepts and techniques is essential for anyone studying algebra and its applications. The ability to find roots of polynomials is not only a valuable mathematical skill but also has practical applications in various fields, such as engineering, physics, and computer science. By mastering these techniques, you can solve a wide range of problems involving polynomial equations.