Polynomial Showdown: Unraveling the Difference (–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – Y5)
The difference between the polynomials (-2x^3y^2 + 4x^2y^3 - 3xy^4) and (6x^4y - 5x^2y^3 - y^5) is calculated by subtracting the second polynomial from the first.
Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. They play a vital role in algebraic equations, representing relationships between different quantities. When dealing with polynomials, it is essential to understand their differences and how they affect the overall expression. In particular, let's dive into the difference between the polynomial (–2x^3y^2 + 4x^2y^3 – 3xy^4) and (6x^4y – 5x^2y^3 – y^5). By examining the contrasts between these two polynomials, we can gain valuable insights into the world of algebra and its applications.
Introduction
Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. They are an essential concept in mathematics and play a crucial role in various fields, including algebra, calculus, and physics. In this article, we will delve into the difference of two polynomials: (–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – y5).
Understanding Polynomials
Before exploring the difference between these two polynomials, let us first gain a clear understanding of what polynomials are. A polynomial is an expression consisting of variables (such as x and y), coefficients (real numbers), and exponents (non-negative integers). The exponents indicate the degree of the variable in the expression.
The First Polynomial: (–2x3y2 + 4x2y3 – 3xy4)
In this polynomial, we have three terms: -2x3y2, 4x2y3, and -3xy4. Each term consists of a coefficient and a variable raised to a certain power. The first term has a coefficient of -2, the variable x raised to the power of 3, and y raised to the power of 2. The second term has a coefficient of 4, the variable x raised to the power of 2, and y raised to the power of 3. Finally, the third term has a coefficient of -3, the variable x raised to the power of 1, and y raised to the power of 4.
The Second Polynomial: (6x4y – 5x2y3 – y5)
The second polynomial consists of three terms as well: 6x4y, -5x2y3, and -y5. The first term has a coefficient of 6, the variable x raised to the power of 4, and y raised to the power of 1. The second term has a coefficient of -5, the variable x raised to the power of 2, and y raised to the power of 3. Lastly, the third term has a coefficient of -1, the variable x raised to the power of 0 (which is always 1), and y raised to the power of 5.
Calculating the Difference
To find the difference between these two polynomials, we need to subtract the second polynomial from the first. Let's perform this calculation:
(–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – y5)
= -2x3y2 + 4x2y3 - 3xy4 - 6x4y + 5x2y3 + y5
Combining Like Terms
Now that we have the expression for the difference of the polynomials, let's combine the like terms. Like terms are terms that have the same variables raised to the same powers. By combining these terms, we can simplify the expression further.
Combining the x3y2 Terms
-2x3y2 - 6x4y = -6x4y - 2x3y2
Combining the x2y3 Terms
4x2y3 + 5x2y3 = 9x2y3
Combining the xy4 Term
-3xy4
Combining the y5 Term
y5
Simplified Expression
After combining the like terms, the simplified expression for the difference of the given polynomials is:
-6x4y - 2x3y2 + 9x2y3 - 3xy4 + y5
Conclusion
In conclusion, the difference between the polynomials (–2x3y2 + 4x2y3 – 3xy4) and (6x4y – 5x2y3 – y5) is -6x4y - 2x3y2 + 9x2y3 - 3xy4 + y5. Understanding polynomials and their operations, such as subtraction, helps us solve various mathematical problems and analyze real-world situations more effectively.
Introduction: Exploring the Difference of Polynomials
Polynomials play a crucial role in mathematics, particularly in algebraic expressions. They consist of terms that are combined using various operations. In this article, we will delve into the concept of polynomial subtraction and explore the difference between two given polynomials.
Identifying the Two Polynomials
The two polynomials in question are: (–2x3y2 + 4x2y3 – 3xy4) and (6x4y – 5x2y3 – y5). To find the difference between these polynomials, we need to subtract the second polynomial from the first.
Understanding the Subtraction Operation Used
Subtracting one polynomial from another involves subtracting corresponding terms. Each term consists of a coefficient and variables raised to certain powers. By subtracting the coefficients and keeping the variables unchanged, we can determine the difference between the two polynomials.
Degree Comparison: Analyzing the highest power of each variable
To compare the degrees of the variables in both polynomials, we look at the highest power of each variable. In the first polynomial, the highest power of x is 3 and the highest power of y is 4. On the other hand, the second polynomial has the highest power of x as 4 and the highest power of y as 5. By comparing these powers, we can determine the differences in degree between the two polynomials.
Coefficient Comparison: Examining the numerical factors of the variables
Next, we examine the coefficients of the variables in both polynomials. In the first polynomial, the coefficients are -2, 4, and -3 for the respective terms. The second polynomial's coefficients are 6, -5, and -1. By comparing these coefficients, we can identify the differences and similarities between the two polynomials.
Variable Similarities: Identifying common variables in both polynomials
Now, let's focus on the variables present in both polynomials. In this case, both polynomials contain x and y. Although the powers of these variables may differ, their presence in both polynomials allows us to identify common terms.
Variable Differences: Isolating variables present in only one polynomial
On the other hand, we also need to isolate variables that appear in only one polynomial. In this case, the first polynomial contains x3y2, x2y3, and xy4, while the second polynomial has x4y and y5. These variables are unique to each polynomial and contribute to the overall difference between them.
Combining Like Terms: Simplifying the polynomial expression after subtraction
By combining like terms, we can simplify the polynomial expression after subtracting the second polynomial from the first. Combining like terms involves adding or subtracting coefficients with the same variables and powers. This step allows us to further analyze and understand the resulting polynomial.
Analyzing the Result: Determining the nature of the resulting polynomial
After simplification, we obtain the resulting polynomial. Analyzing this polynomial helps us determine its nature and specific properties. For example, we can determine whether it is a binomial, trinomial, or any other type of polynomial. Additionally, we can identify its degree and coefficients to gain a deeper understanding of the result.
Conclusion: Summarizing the differences and implications of the polynomials' subtraction
In conclusion, exploring the difference between polynomials involves various steps. We begin by identifying the two polynomials and understanding the subtraction operation used. Then, we compare the degrees and coefficients of variables, identify similarities and differences, and combine like terms to simplify the expression. Finally, we analyze the resulting polynomial to draw meaningful conclusions about its nature. By following these steps, we can gain a comprehensive understanding of the difference between polynomials and its implications in algebraic expressions.
In order to understand the difference between two polynomials, let's analyze the expression (–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – y5) step by step:
- First, let's focus on the coefficients of the terms in the polynomial. The terms in the first polynomial are (-2x^3y^2), (4x^2y^3), and (-3xy^4), while the terms in the second polynomial are (6x^4y), (-5x^2y^3), and (-y^5).
- Next, let's compare the exponents of the variables present in each term. In the first polynomial, we have x raised to the power of 3, 2, and 1, and y raised to the power of 2, 3, and 4. In the second polynomial, we have x raised to the power of 4 and 2, and y raised to the power of 1, 3, and 5.
- Now, let's subtract the corresponding terms from each other. Starting with the x terms, we have -2x^3 - 0x^4 (since there is no corresponding term in the second polynomial) = -2x^3. Moving on to the x^2 terms, we have 4x^2 - (-5x^2) = 4x^2 + 5x^2 = 9x^2. Continuing this process for the remaining terms, we get -2x^3y^2 + 9x^2y^3 - (-y^5) = -2x^3y^2 + 9x^2y^3 + y^5.
- Finally, let's simplify the expression we obtained. The difference between the two polynomials is -2x^3y^2 + 9x^2y^3 + y^5.
In conclusion, the difference between the polynomials (–2x^3y^2 + 4x^2y^3 – 3xy^4) and (6x^4y – 5x^2y^3 – y^5) is represented by the polynomial -2x^3y^2 + 9x^2y^3 + y^5.
Thank you for visiting our blog and taking the time to read about the difference of polynomials. In this article, we will discuss the difference between two given polynomials: (–2x3y2 + 4x2y3 – 3xy4) and (6x4y – 5x2y3 – y5). By understanding and analyzing their differences, we can gain important insights into polynomial operations and their applications.
To find the difference of the two polynomials, we need to subtract the second polynomial from the first one. This is done by changing the sign of each term in the second polynomial and then combining like terms. Let's break down the process step by step to ensure a clear understanding.
First, let's change the sign of each term in the second polynomial. This means that the positive coefficients become negative, and the negative coefficients become positive. By doing so, the second polynomial becomes (-6x4y + 5x2y3 + y5). Now, we can combine like terms by adding or subtracting the coefficients of the same variables.
For example, the terms -2x3y2 and -6x4y have different variables and exponents, so they cannot be combined. However, the terms 4x2y3 and 5x2y3 have the same variables (x and y) and the same exponent (2 and 3), so we can combine them by adding their coefficients. The result is -2x3y2 + 9x2y3.
Continuing this process, we combine the terms -3xy4 and y5, which gives us -3xy4 - y5. Finally, we have the simplified form of the difference of the polynomials as -2x3y2 + 9x2y3 - 3xy4 - y5.
In conclusion, understanding the difference of polynomials is an important concept in algebra. By following the steps mentioned above and carefully combining like terms, we can find the simplified form of the difference between two polynomials. This knowledge is valuable for solving equations, simplifying expressions, and analyzing real-world problems. We hope this article has provided you with a clear understanding of finding the difference of polynomials. Thank you once again for visiting our blog!
What Is The Difference Of The Polynomials?
Explanation:
When subtracting polynomials, we need to combine like terms and simplify the expression. Let's subtract the polynomial (–2x³y² + 4x²y³ – 3xy⁴) from the polynomial (6x⁴y – 5x²y³ – y⁵).
Step 1:
Write down both polynomials:
- Polynomial 1: –2x³y² + 4x²y³ – 3xy⁴
- Polynomial 2: 6x⁴y – 5x²y³ – y⁵
Step 2:
Since we are subtracting the second polynomial from the first one, we change the signs of all terms in the second polynomial:
- Polynomial 1: –2x³y² + 4x²y³ – 3xy⁴
- Polynomial 2: –6x⁴y + 5x²y³ + y⁵
Step 3:
Combine like terms:
- -2x³y² - (-6x⁴y) = -2x³y² + 6x⁴y = 6x⁴y - 2x³y²
- 4x²y³ - 5x²y³ = -x²y³
- -3xy⁴ - y⁵ = -3xy⁴ - y⁵
Step 4:
Write the simplified expression:
- 6x⁴y - 2x³y² - x²y³ - 3xy⁴ - y⁵
Therefore, the difference of the polynomials (–2x³y² + 4x²y³ – 3xy⁴) and (6x⁴y – 5x²y³ – y⁵) is 6x⁴y - 2x³y² - x²y³ - 3xy⁴ - y⁵.