How to Factorize the Given Polynomial: -m-n 36x^2 49ny^2 - 49my^2

In this article, we will explore how to factorize the given polynomial -m-n 36x^2 49ny^2 - 49my^2. We will delve into the steps required to fully factorize the expression and determine its prime factors. By the end of this guide, you will understand the process and be able to factorize similar polynomials.

Introduction

The polynomial in question is -m-n 36x^2 49ny^2 - 49my^2. We will assume that M is a sloppy notation for m, as the expression yields no interesting result otherwise. Our goal is to factorize this polynomial into a product of irreducible polynomials.

Step-by-Step Factorization

Let's begin with the given polynomial:

-m-n 36x^2 49ny^2 - 49my^2

Step 1: Identifying the Greatest Common Divisor (GCD)

We notice that the terms 49ny^2 - 49my^2 share a common factor of 49y^2. We can factor this out:

49ny^2 - 49my^2 49y^2(n - m)

Substituting this back into the original polynomial:

36x^2(n - m) 49y^2(n - m)

Step 2: Factoring Out the Common Factor

Both terms now share a common factor of (n - m). We can factor this out:

(n - m)(36x^2 49y^2)

Step 3: Factoring the Quadratic Expression

Next, we need to factor the quadratic expression 36x^2 49y^2. To do this, we will use the identity for the sum of squares:

a^2 b^2 (a bi)(a - bi)

In our case, we can rewrite the expression as follows:

36x^2 49y^2 (6x)^2 (7y)^2

Using the identity:

(6x)^2 (7y)^2 (6x 7yi)(6x - 7yi)

Final Factorization

Combining all the steps, we get:

-m-n 36x^2 49ny^2 - 49my^2 (n - m)(6x 7yi)(6x - 7yi)

Irreducibility of the Factors

To determine if the factors are irreducible, we need to consider the degrees of the polynomials and the coefficients:

1. 6x 7yi and 6x - 7yi

Both factors 6x 7yi and 6x - 7yi are of degree 1 in x and y. Since the coefficients (6 and 7) are coprime, these factors are irreducible.

2. n - m

The factor n - m is a linear factor and is irreducible in the context of polynomials with integer coefficients.

Conclusion

In this article, we have successfully factorized the given polynomial -m-n 36x^2 49ny^2 - 49my^2. The final factorization is: