Find Polynomial M For Equation Identity M + (5x^2 - 2xy) = 6x^2 + 9xy - Y^2

Jul 11, 2025 by ADMIN 76 views

Find the Polynomial M that Satisfies the Identity

In the realm of algebra, the pursuit of identities often leads us to explore the intricate relationships between polynomials. An identity, in mathematical terms, is an equation that holds true for all possible values of its variables. When dealing with polynomials, this means finding expressions that, when substituted, make the equation a universal truth. In this article, we embark on a journey to uncover the polynomial M that, when substituted into a given equation, transforms it into an identity. Specifically, we aim to solve for M in the equation:

$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

This endeavor requires a firm grasp of algebraic manipulation and a keen eye for simplification. We will delve into the techniques of polynomial arithmetic, focusing on addition, subtraction, and the concept of like terms. By the end of this exploration, you will not only understand how to find the polynomial M but also appreciate the underlying principles that govern polynomial identities. Let's embark on this mathematical quest, unraveling the mystery of polynomial M and its role in creating an identity.

Understanding Polynomial Identities

Before we dive into the solution, it's crucial to understand the concept of a polynomial identity. A polynomial identity is an equation involving polynomials that is true for all possible values of the variables. For instance, the equation $(x + y)^2 = x^2 + 2xy + y^2$ is a well-known identity. No matter what values you substitute for x and y, the equation will always hold true. In our case, we're looking for a polynomial M that, when added to $(5x^2 - 2xy)$ , will always equal $(6x^2 + 9xy - y^2)$ .

To find this polynomial M, we'll employ the principles of algebraic manipulation. The core idea is to isolate M on one side of the equation. This involves subtracting the polynomial $(5x^2 - 2xy)$ from both sides of the equation. This process hinges on the properties of equality, which state that performing the same operation on both sides of an equation maintains the equality. By carefully applying these principles, we can systematically unravel the equation and reveal the identity. The polynomial M will then become clear, and we'll be able to verify its correctness by substituting it back into the original equation.

Solving for M: A Step-by-Step Approach

Let's begin by rewriting the equation:

$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

Our goal is to isolate M. To do this, we need to subtract the polynomial $(5x^2 - 2xy)$ from both sides of the equation. This ensures that we maintain the equality while moving terms around. The process of subtracting a polynomial involves changing the sign of each term within the polynomial and then combining like terms. Like terms are those that have the same variables raised to the same powers. For example, $5x^2$ and $6x^2$ are like terms, while $2xy$ and $9xy$ are also like terms. Subtracting unlike terms is not possible; we can only indicate the subtraction.

Subtracting $(5x^2 - 2xy)$ from both sides, we get:

$M + (5x^2 - 2xy) - (5x^2 - 2xy) = (6x^2 + 9xy - y^2) - (5x^2 - 2xy)$

On the left side, the $(5x^2 - 2xy)$ terms cancel out, leaving us with just M:

$M = (6x^2 + 9xy - y^2) - (5x^2 - 2xy)$

Now, we need to simplify the right side. This involves distributing the negative sign to each term inside the second parenthesis and then combining like terms:

$M = 6x^2 + 9xy - y^2 - 5x^2 + 2xy$

Next, we identify and combine the like terms:

$x^2$ terms: $6x^2 - 5x^2 = x^2$
$xy$ terms: $9xy + 2xy = 11xy$
$y^2$ terms: $-y^2$ (there's only one $y^2$ term)

Combining these, we get:

$M = x^2 + 11xy - y^2$

Therefore, the polynomial M that makes the equation an identity is $x^2 + 11xy - y^2$ .

Verification: Confirming the Identity

To ensure that our solution is correct, we need to verify that substituting $M = x^2 + 11xy - y^2$ back into the original equation indeed creates an identity. This means that the left-hand side of the equation should equal the right-hand side for all values of x and y. This is a crucial step in the process of solving mathematical problems, as it provides a check against errors in our calculations and reasoning.

Let's substitute our solution for M into the original equation:

$(x^2 + 11xy - y^2) + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

Now, we simplify the left-hand side by combining like terms:

$x^2$ terms: $x^2 + 5x^2 = 6x^2$
$xy$ terms: $11xy - 2xy = 9xy$
$y^2$ terms: $-y^2$

So, the left-hand side becomes:

$6x^2 + 9xy - y^2$

This is exactly the same as the right-hand side of the original equation. Therefore, our solution for polynomial M is indeed correct. This verification step not only confirms our answer but also reinforces the concept of an identity – an equation that holds true for all possible values of the variables.

Conclusion: The Power of Polynomial Manipulation

In this exploration, we successfully found the polynomial M, which is $x^2 + 11xy - y^2$ , that satisfies the equation

$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

and transforms it into an identity. We achieved this by applying fundamental algebraic principles, such as isolating variables and combining like terms. The process of verifying our solution by substitution further solidified our understanding of polynomial identities.

This exercise highlights the power and elegance of polynomial manipulation in solving algebraic problems. By mastering these techniques, we can tackle more complex equations and gain a deeper appreciation for the structure and properties of polynomials. The ability to manipulate polynomials is a cornerstone of algebra, with applications extending to various fields, including calculus, linear algebra, and computer science. This journey into finding polynomial M serves as a stepping stone towards further mathematical explorations and problem-solving endeavors.

Problem:

Find a polynomial M such that the equation

$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

is an identity. In other words, find polynomial M that, when substituted into the equation, makes the equation true for all values of x and y. This problem falls under the domain of algebra, specifically the manipulation of polynomials. Understanding how to add, subtract, and simplify polynomials is crucial for solving this type of problem. Moreover, the concept of an identity is central to the solution. An identity is an equation that holds true for all values of the variables involved. Our goal is to find a polynomial M that ensures the given equation is an identity.

To approach this problem, we need to isolate M on one side of the equation. This can be achieved by performing algebraic operations on both sides of the equation, ensuring that the equality is maintained. The key step involves subtracting the polynomial $(5x^2 - 2xy)$ from both sides of the equation. This will effectively eliminate the polynomial from the left-hand side, leaving us with M isolated. Once we have isolated M, we will need to simplify the expression on the right-hand side by combining like terms. Like terms are terms that have the same variables raised to the same powers. For example, $5x^2$ and $6x^2$ are like terms, while $2xy$ and $9xy$ are also like terms. Combining like terms involves adding or subtracting their coefficients.

Solution:

To find the polynomial M, we will follow these steps:

Start with the given equation:

$M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$
Subtract $(5x^2 - 2xy)$ from both sides of the equation to isolate M:

$M = (6x^2 + 9xy - y^2) - (5x^2 - 2xy)$
Distribute the negative sign to the terms inside the second parenthesis:

$M = 6x^2 + 9xy - y^2 - 5x^2 + 2xy$
Combine like terms:
- Combine the $x^2$ terms: $6x^2 - 5x^2 = x^2$
- Combine the $xy$ terms: $9xy + 2xy = 11xy$
- The $y^2$ term remains as $-y^2$ since there are no other $y^2$ terms.
Write the simplified expression for M:

$M = x^2 + 11xy - y^2$

Therefore, the polynomial M that makes the equation an identity is $x^2 + 11xy - y^2$ .

Verification:

To verify our solution, we substitute $M = x^2 + 11xy - y^2$ back into the original equation:

$(x^2 + 11xy - y^2) + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$

Combine like terms on the left-hand side:

Combine the $x^2$ terms: $x^2 + 5x^2 = 6x^2$
Combine the $xy$ terms: $11xy - 2xy = 9xy$
The $y^2$ term remains as $-y^2$

So, the left-hand side becomes:

$6x^2 + 9xy - y^2$

This is equal to the right-hand side of the original equation, which confirms that our solution is correct. Thus, the polynomial M that satisfies the equation is indeed $x^2 + 11xy - y^2$ . This verification step is crucial as it provides assurance that the solution obtained is accurate and that no errors were made during the algebraic manipulations. The ability to verify solutions is a hallmark of strong problem-solving skills in mathematics.

Conclusion:

We have successfully found the polynomial M that makes the given equation an identity. By isolating M, simplifying the expression, and verifying the solution, we have demonstrated a systematic approach to solving this type of algebraic problem. The polynomial M, which is $x^2 + 11xy - y^2$ , ensures that the equation $M + (5x^2 - 2xy) = 6x^2 + 9xy - y^2$ holds true for all values of x and y. This problem underscores the importance of understanding polynomial arithmetic and the concept of identities in algebra. The ability to manipulate polynomials and solve for unknown expressions is a fundamental skill in mathematics, with applications in various fields such as engineering, physics, and computer science. Mastering these skills opens doors to more advanced mathematical concepts and problem-solving techniques.

In summary, finding the polynomial M involved the following steps: isolating M by subtracting the given polynomial from both sides of the equation, simplifying the resulting expression by combining like terms, and verifying the solution by substituting it back into the original equation. This systematic approach is essential for solving algebraic problems accurately and efficiently. The polynomial M that satisfies the equation is $x^2 + 11xy - y^2$ , and it makes the equation an identity, meaning it holds true for all possible values of x and y. This concludes our exploration of finding the polynomial M that satisfies the given equation and makes it an identity. The principles and techniques used here are applicable to a wide range of algebraic problems, highlighting the importance of mastering fundamental algebraic concepts.