Proving The Relationship Between Factors Remainders And Coefficients In Quadratic Equations

Introduction

In the realm of algebra, the interplay between factors, remainders, and coefficients in polynomial equations reveals fascinating relationships. This article delves into a specific scenario involving quadratic equations, exploring the connection between a factor x - p of a quadratic expression and the remainder q obtained when another related quadratic expression is divided by x + p. Our central objective is to demonstrate that, given certain conditions, the parameter p can be expressed as the square root of the quotient of q and a - 1, where a is a coefficient in the quadratic expressions, and a is not equal to 1. This exploration will not only enhance your understanding of polynomial algebra but also showcase the power of algebraic manipulation and logical deduction in establishing mathematical truths.

Understanding the Factor Theorem and Remainder Theorem

At the heart of our investigation lie two fundamental theorems: the Factor Theorem and the Remainder Theorem. The Factor Theorem states that for a polynomial f(x), if f(p) = 0, then x - p is a factor of f(x). Conversely, if x - p is a factor of f(x), then f(p) = 0. This theorem provides a direct link between the roots of a polynomial and its factors. The Remainder Theorem, on the other hand, states that when a polynomial f(x) is divided by x - p, the remainder is f(p). This theorem allows us to determine the remainder of a polynomial division without actually performing the long division process. These theorems are crucial tools in analyzing polynomial behavior and form the basis for many algebraic techniques.

Setting Up the Problem

Let's consider the given scenario. We have two quadratic expressions: x^2 + bx - c and ax^2 - bx - c. We are told that x - p is a factor of the first quadratic, x^2 + bx - c. This implies, by the Factor Theorem, that substituting x = p into the quadratic will result in zero. Mathematically, this can be written as:

p^2 + bp - c = 0

Furthermore, we are given that when ax^2 - bx - c is divided by x + p, the remainder is q. According to the Remainder Theorem, this means that substituting x = -p into the quadratic ax^2 - bx - c will yield the remainder q. This can be expressed as:

a(-p)^2 - b(-p) - c = q

Simplifying this equation, we get:

ap^2 + bp - c = q

Our goal is to use these two equations to demonstrate that p = √(q / (a - 1)), given that a ≠ 1. The strategy involves manipulating these equations to eliminate the variable b and express p in terms of q and a. This will require careful algebraic manipulation and a keen understanding of equation solving techniques.

Derivation of the Relationship

Leveraging the Factor Theorem

As previously established, the fact that x - p is a factor of x^2 + bx - c allows us to write the equation:

p^2 + bp - c = 0

This equation is a direct consequence of the Factor Theorem, which links the roots of a polynomial to its factors. The substitution of x = p into the quadratic expression results in zero, indicating that p is a root of the equation x^2 + bx - c = 0. This equation forms the foundation for our subsequent algebraic manipulations. We can rearrange this equation to express bp in terms of p^2 and c:

bp = c - p^2

This expression will be crucial in eliminating the variable b from our equations, allowing us to focus on the relationship between p, q, and a.

Applying the Remainder Theorem

We also know that when ax^2 - bx - c is divided by x + p, the remainder is q. This leads to the equation:

ap^2 + bp - c = q

This equation arises from the Remainder Theorem, which states that substituting x = -p into the quadratic expression yields the remainder. This equation provides another key piece of information in our puzzle, linking the coefficients a, b, and c to the remainder q and the parameter p. Now, we can substitute the expression for bp that we derived earlier (bp = c - p^2) into this equation:

ap^2 + (c - p^2) - c = q

This substitution is a critical step in our derivation, as it eliminates the variable b and allows us to work with an equation involving only a, p, c, and q.

Simplifying and Isolating p

Now, let's simplify the equation obtained after the substitution:

ap^2 + c - p^2 - c = q

Notice that the c terms cancel out, leaving us with:

ap^2 - p^2 = q

This simplification brings us closer to our goal of expressing p in terms of q and a. We can factor out p^2 from the left side of the equation:

p^2(a - 1) = q

This factorization highlights the relationship between p^2, q, and the difference a - 1. Now, we can isolate p^2 by dividing both sides of the equation by (a - 1):

p^2 = q / (a - 1)

This step is valid only if a - 1 ≠ 0, which implies that a ≠ 1, a condition that was given in the problem statement. Finally, to find p, we take the square root of both sides of the equation:

p = ±√(q / (a - 1))

Thus, we have successfully demonstrated that p is equal to the square root of q divided by a - 1, under the condition that a ≠ 1.

Conclusion

In this exploration, we have successfully proven that if x - p is a factor of x^2 + bx - c and ax^2 - bx - c has a remainder of q when divided by x + p, then p = √(q / (a - 1)), provided that a ≠ 1. This result showcases the elegant interplay between the Factor Theorem and the Remainder Theorem, and how they can be used to derive relationships between the coefficients, roots, and remainders of polynomial equations. The derivation involved careful algebraic manipulation, substitution, and simplification, highlighting the importance of these techniques in solving mathematical problems. This exploration not only provides a specific result but also reinforces the understanding of fundamental algebraic principles that are applicable in a wide range of mathematical contexts.

By understanding these relationships, we gain a deeper appreciation for the structure and behavior of polynomials, which are fundamental building blocks in various fields of mathematics, science, and engineering. This result serves as a testament to the power of algebraic reasoning and the beauty of mathematical connections.

This exploration underscores the significance of mastering fundamental theorems and algebraic techniques in unraveling complex mathematical relationships. The ability to connect seemingly disparate concepts, such as factors, remainders, and coefficients, is a hallmark of mathematical proficiency. As we continue our journey in mathematics, these skills will serve as invaluable tools in tackling more challenging problems and exploring new mathematical frontiers.