Polynomial Subtraction Analysis Determining The Difference Between 6x⁶ - X³y⁴ - 5xy⁵ And 4x⁵y + 2x² - 5xy⁵

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In the realm of mathematics, polynomials stand as fundamental building blocks, underpinning numerous concepts and applications. Polynomials, expressions comprising variables and coefficients, intertwined through addition, subtraction, and multiplication, dictate the landscape of algebraic equations and functions. Among the myriad operations applicable to polynomials, subtraction emerges as a cornerstone, enabling us to dissect and compare these expressions with precision. In this article, we delve into the intricacies of polynomial subtraction, focusing on the specific case of the polynomials 6x⁶ - x³y⁴ - 5xy⁵ and 4x⁵y + 2x² - 5xy⁵. Our primary objective is to unravel the true nature of the simplified difference between these polynomials, scrutinizing its terms and degree to ascertain the correct characterization.

Understanding Polynomial Subtraction

Before we embark on the expedition of subtracting the given polynomials, it's crucial to lay a solid foundation of understanding regarding polynomial subtraction itself. Polynomial subtraction is, at its core, the process of finding the difference between two or more polynomials. This operation involves combining like terms, those terms sharing the same variables raised to the same powers, while adhering to the fundamental principles of algebraic manipulation. The essence of polynomial subtraction lies in distributing the negative sign across the terms of the polynomial being subtracted, effectively transforming subtraction into addition of the additive inverse. This transformation paves the way for combining like terms and simplifying the expression to its most concise form. The result of polynomial subtraction is another polynomial, whose characteristics, such as the number of terms and degree, are determined by the interplay of the original polynomials.

Step-by-Step Subtraction Process

To concretize the concept of polynomial subtraction, let's outline the step-by-step process:

  1. Write the polynomials: Begin by expressing the polynomials to be subtracted in a clear and organized manner. This sets the stage for the subsequent steps.
  2. Distribute the negative sign: Distribute the negative sign to each term of the polynomial being subtracted. This crucial step transforms subtraction into addition of the additive inverse, enabling us to combine like terms effectively.
  3. Identify like terms: Scrutinize the polynomials, pinpointing terms that share the same variables raised to the same powers. These are the terms that can be combined in the next step.
  4. Combine like terms: Employ the rules of addition and subtraction to combine the coefficients of like terms. This step consolidates the expression, bringing it closer to its simplified form.
  5. Simplify: Examine the resulting expression, ensuring that it is in its most concise form. This may involve combining additional like terms or rearranging the expression for clarity.

By adhering to these steps, we can confidently navigate the realm of polynomial subtraction, arriving at accurate and simplified results.

Subtracting the Polynomials: A Detailed Walkthrough

Now, let's turn our attention to the specific polynomials at hand: 6x⁶ - x³y⁴ - 5xy⁵ and 4x⁵y + 2x² - 5xy⁵. Our mission is to subtract the second polynomial from the first, meticulously following the steps outlined earlier.

  1. Write the polynomials: We begin by expressing the polynomials to be subtracted: (6x⁶ - x³y⁴ - 5xy⁵) - (4x⁵y + 2x² - 5xy⁵)
  2. Distribute the negative sign: We distribute the negative sign to each term of the second polynomial: 6x⁶ - x³y⁴ - 5xy⁵ - 4x⁵y - 2x² + 5xy⁵
  3. Identify like terms: We carefully scrutinize the expression, seeking out terms that share the same variables raised to the same powers. In this case, we observe that -5xy⁵ and +5xy⁵ are like terms.
  4. Combine like terms: We combine the like terms: 6x⁶ - x³y⁴ - 4x⁵y - 2x² + (-5xy⁵ + 5xy⁵) This simplifies to: 6x⁶ - x³y⁴ - 4x⁵y - 2x²
  5. Simplify: We examine the resulting expression, ensuring that it is in its most concise form. In this instance, the expression is already simplified.

Therefore, the completely simplified difference of the polynomials 6x⁶ - x³y⁴ - 5xy⁵ and 4x⁵y + 2x² - 5xy⁵ is 6x⁶ - x³y⁴ - 4x⁵y - 2x².

Analyzing the Result: Terms and Degree

With the simplified difference in hand, we now turn our attention to analyzing its characteristics, specifically the number of terms and the degree. These attributes provide valuable insights into the nature of the resulting polynomial.

Number of Terms

The number of terms in a polynomial is simply the count of individual expressions separated by addition or subtraction signs. In our simplified difference, 6x⁶ - x³y⁴ - 4x⁵y - 2x², we can readily identify four distinct terms: 6x⁶, -x³y⁴, -4x⁵y, and -2x². Thus, the difference has 4 terms.

Degree of the Polynomial

The degree of a polynomial is a measure of its complexity, determined by the highest sum of the exponents of variables within a single term. To ascertain the degree of our simplified difference, we must examine each term individually:

  • 6x⁶: The degree of this term is 6, as the exponent of the variable 'x' is 6.
  • -x³y⁴: The degree of this term is 7, obtained by summing the exponents of 'x' (3) and 'y' (4).
  • -4x⁵y: The degree of this term is 6, calculated by adding the exponents of 'x' (5) and 'y' (1).
  • -2x²: The degree of this term is 2, as the exponent of 'x' is 2.

Among these terms, -x³y⁴ exhibits the highest degree, 7. Consequently, the degree of the entire polynomial, the simplified difference, is 7.

Conclusion: Identifying the Correct Statement

Based on our meticulous analysis, we have determined that the completely simplified difference of the polynomials 6x⁶ - x³y⁴ - 5xy⁵ and 4x⁵y + 2x² - 5xy⁵ is 6x⁶ - x³y⁴ - 4x⁵y - 2x². This difference comprises four terms and possesses a degree of 7.

Now, let's revisit the options presented in the original question:

  • A. The difference has 3 terms and a degree of 6.
  • B. The difference has 4 terms and a degree of 6.

Comparing our findings with the provided options, we can confidently conclude that neither option accurately describes the simplified difference. The difference has 4 terms and a degree of 7, which contradicts both options.

In summary, our exploration of polynomial subtraction has unveiled the intricacies of this fundamental operation, allowing us to dissect and compare polynomial expressions with precision. By meticulously following the steps of polynomial subtraction and carefully analyzing the resulting expression, we have arrived at a comprehensive understanding of the difference between the given polynomials, ultimately leading us to the correct characterization of its terms and degree. This exercise underscores the importance of a solid foundation in algebraic principles and the power of systematic analysis in unraveling mathematical complexities.