Simplified Difference Of Polynomials $a^3 B+9 A^2 B^2-4 A B^5$ And $a^3 B-3 A^2 B^2+a B^5$

In the realm of mathematics, polynomials stand as fundamental expressions, and understanding their operations, especially subtraction, is crucial. This article delves into the intricacies of finding the simplified difference between two specific polynomials: $a^3 b+9 a^2 b^2-4 a b^5$ and $a^3 b-3 a^2 b^2+a b^5$. We will meticulously walk through the simplification process, identify the resulting polynomial's characteristics, and determine the correct answer from the given options.

Demystifying Polynomial Subtraction

Polynomial subtraction involves combining like terms, which are terms with the same variables raised to the same powers. The core concept is to distribute the negative sign across the second polynomial and then add the resulting terms to the first polynomial. This process ensures that we accurately account for the subtraction operation. Mastering polynomial subtraction is a cornerstone of algebraic manipulation, enabling us to solve a wide array of mathematical problems. The ability to confidently subtract polynomials opens doors to more advanced concepts such as polynomial division, factoring, and solving polynomial equations. Furthermore, polynomial subtraction has practical applications in various fields, including physics, engineering, and computer science, where mathematical models often involve polynomial expressions.

Before we embark on the specific problem at hand, let's recap the fundamental principles of polynomial subtraction. Remember that subtracting a polynomial is equivalent to adding its additive inverse. In other words, we change the sign of each term in the second polynomial and then combine like terms. This seemingly simple process requires careful attention to detail to avoid errors. For instance, consider subtracting the polynomial $2x^2 - 3x + 1$ from $5x^2 + x - 4$. We change the signs of the terms in the second polynomial to get $-2x^2 + 3x - 1$, and then we add this to the first polynomial: $(5x^2 + x - 4) + (-2x^2 + 3x - 1)$. Combining like terms, we have $(5x^2 - 2x^2) + (x + 3x) + (-4 - 1) = 3x^2 + 4x - 5$. This example highlights the importance of correctly distributing the negative sign and combining like terms with precision.

Now, let's transition to the problem at hand. We are tasked with finding the simplified difference between the polynomials $a^3 b+9 a^2 b^2-4 a b^5$ and $a^3 b-3 a^2 b^2+a b^5$. The first step is to rewrite the subtraction as addition of the additive inverse: $(a^3 b+9 a^2 b^2-4 a b^5) - (a^3 b-3 a^2 b^2+a b^5) = (a^3 b+9 a^2 b^2-4 a b^5) + (-a^3 b+3 a^2 b^2-a b^5)$. Next, we identify and combine like terms. We have $a^3 b$ and $-a^3 b$, $9 a^2 b^2$ and $3 a^2 b^2$, and $-4 a b^5$ and $-a b^5$. Combining these terms gives us $(a^3 b - a^3 b) + (9 a^2 b^2 + 3 a^2 b^2) + (-4 a b^5 - a b^5)$. Simplifying each group of like terms, we get $0 a^3 b + 12 a^2 b^2 - 5 a b^5$. Thus, the simplified difference is $12 a^2 b^2 - 5 a b^5$.

Step-by-Step Simplification

To find the simplified difference of the polynomials $a^3 b+9 a^2 b^2-4 a b^5$ and $a^3 b-3 a^2 b^2+a b^5$, we follow a meticulous step-by-step process. This approach ensures accuracy and clarity in our solution.

1. Write the subtraction: $(a^3 b+9 a^2 b^2-4 a b^5) - (a^3 b-3 a^2 b^2+a b^5)$
2. Distribute the negative sign: This is a crucial step where we change the signs of each term in the second polynomial. The expression becomes: $a^3 b+9 a^2 b^2-4 a b^5 - a^3 b+3 a^2 b^2-a b^5$
3. Identify like terms: Like terms are those that have the same variables raised to the same powers. In this case, we have:
   • $a^3 b$ and $-a^3 b$
   • $9 a^2 b^2$ and $3 a^2 b^2$
   • $-4 a b^5$ and $-a b^5$
4. Combine like terms: This involves adding or subtracting the coefficients of the like terms:
   • $(a^3 b - a^3 b) = 0$
   • $(9 a^2 b^2 + 3 a^2 b^2) = 12 a^2 b^2$
   • $(-4 a b^5 - a b^5) = -5 a b^5$
5. Write the simplified polynomial: Combining the results, we get $12 a^2 b^2 - 5 a b^5$

This step-by-step breakdown clarifies the process of polynomial subtraction, highlighting the importance of distributing the negative sign and combining like terms accurately. By following these steps, we can confidently simplify polynomial expressions and arrive at the correct solution.

Analyzing the Resulting Polynomial

Now that we have the simplified difference, $12 a^2 b^2 - 5 a b^5$, we need to analyze its characteristics to determine the correct answer from the given options. The key aspects to consider are the number of terms and the degree of the polynomial. Understanding these concepts is crucial for classifying polynomials and performing further operations on them.

The number of terms in a polynomial determines its classification. A polynomial with one term is called a monomial, with two terms a binomial, with three terms a trinomial, and so on. In our simplified difference, $12 a^2 b^2 - 5 a b^5$, we have two terms: $12 a^2 b^2$ and $-5 a b^5$. Therefore, the simplified difference is a binomial.

The degree of a term in a polynomial is the sum of the exponents of the variables in that term. For example, in the term $12 a^2 b^2$, the degree is $2 + 2 = 4$, and in the term $-5 a b^5$, the degree is $1 + 5 = 6$. The degree of the polynomial itself is the highest degree among its terms. In this case, the highest degree is 6, which comes from the term $-5 a b^5$.

Therefore, the simplified difference $12 a^2 b^2 - 5 a b^5$ is a binomial with a degree of 6. This analysis aligns with one of the provided options, which we will identify in the next section.

Identifying the Correct Answer

Based on our simplification and analysis, we've determined that the difference between the polynomials $a^3 b+9 a^2 b^2-4 a b^5$ and $a^3 b-3 a^2 b^2+a b^5$ is the binomial $12 a^2 b^2 - 5 a b^5$, which has a degree of 6. Now, let's examine the given options:

A. The difference is a binomial with a degree of 5. B. The difference is a binomial with a degree of 6. C. [This option is not provided in the original context]

Comparing our result with the options, we can clearly see that option B accurately describes the simplified difference. It correctly identifies the result as a binomial (two terms) with a degree of 6, matching our findings.

Therefore, the correct answer is B. The difference is a binomial with a degree of 6. This conclusion is reached through a systematic process of polynomial subtraction, simplification, and analysis of the resulting expression.

Conclusion: Mastering Polynomial Subtraction

In this comprehensive guide, we have meticulously explored the process of finding the simplified difference between the polynomials $a^3 b+9 a^2 b^2-4 a b^5$ and $a^3 b-3 a^2 b^2+a b^5$. We began by understanding the fundamental principles of polynomial subtraction, emphasizing the importance of distributing the negative sign and combining like terms. We then executed a step-by-step simplification, arriving at the binomial $12 a^2 b^2 - 5 a b^5$.

Further analysis revealed that this binomial has a degree of 6, leading us to the correct answer: B. The difference is a binomial with a degree of 6. This exercise underscores the significance of a systematic approach to polynomial operations, ensuring accuracy and clarity in our solutions.

Polynomial subtraction, like other algebraic operations, forms the bedrock of more advanced mathematical concepts. By mastering these fundamental skills, we equip ourselves to tackle complex problems in various fields, from engineering and physics to computer science and economics. Continuous practice and a deep understanding of the underlying principles are key to unlocking the full potential of polynomial algebra.