Which Is True About The Completely Simplified Difference Of The Polynomials A 3 B + 9 A 2 B 2 − 4 A B 5 A^3 B + 9a^2 B^2 - 4ab^5 A 3 B + 9 A 2 B 2 − 4 A B 5 And A 3 B − 3 A 2 B 2 + A B 5 A^3 B - 3a^2 B^2 + Ab^5 A 3 B − 3 A 2 B 2 + A B 5 ?A. The Difference Is A Binomial With A Degree Of 5.B. The Difference Is A Binomial With A Degree Of

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Introduction

Polynomials are a fundamental concept in algebra, and understanding how to simplify their differences is crucial for solving various mathematical problems. In this article, we will delve into the world of polynomial differences, focusing on the completely simplified difference of two given polynomials. We will explore the properties of polynomial differences, identify the degree of the resulting polynomial, and provide a step-by-step guide on how to simplify the difference.

Understanding Polynomial Differences

A polynomial difference is the result of subtracting one polynomial from another. When simplifying polynomial differences, we need to apply the rules of polynomial arithmetic, which include the distributive property, the commutative property, and the associative property.

The Given Polynomials

Let's consider the two given polynomials:

  1. a3b+9a2b2−4ab5a^3 b + 9a^2 b^2 - 4ab^5
  2. a3b−3a2b2+ab5a^3 b - 3a^2 b^2 + ab^5

Step 1: Subtract the Second Polynomial from the First

To find the difference between the two polynomials, we need to subtract the second polynomial from the first. This involves subtracting corresponding terms:

(a3b+9a2b2−4ab5)−(a3b−3a2b2+ab5)(a^3 b + 9a^2 b^2 - 4ab^5) - (a^3 b - 3a^2 b^2 + ab^5)

Step 2: Simplify the Expression

Now, let's simplify the expression by combining like terms:

a3b−a3b+9a2b2+3a2b2−4ab5−ab5a^3 b - a^3 b + 9a^2 b^2 + 3a^2 b^2 - 4ab^5 - ab^5

Step 3: Combine Like Terms

Combining like terms, we get:

12a2b2−5ab512a^2 b^2 - 5ab^5

The Simplified Difference

The simplified difference of the two polynomials is:

12a2b2−5ab512a^2 b^2 - 5ab^5

Analyzing the Degree of the Resulting Polynomial

Now that we have simplified the difference, let's analyze the degree of the resulting polynomial. The degree of a polynomial is the highest power of the variable (in this case, bb) with a non-zero coefficient.

In the simplified difference, the highest power of bb is 5, which is the degree of the polynomial.

Conclusion

In conclusion, the completely simplified difference of the two given polynomials is 12a2b2−5ab512a^2 b^2 - 5ab^5. The degree of the resulting polynomial is 5, which is the highest power of the variable bb with a non-zero coefficient.

Answer to the Question

Based on our analysis, the correct answer to the question is:

A. The difference is a binomial with a degree of 5.

Final Thoughts

Q: What is the difference between a polynomial and a binomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A binomial, on the other hand, is a polynomial with exactly two terms.

Q: How do I simplify a polynomial difference?

A: To simplify a polynomial difference, you need to subtract the second polynomial from the first, combining like terms and applying the rules of polynomial arithmetic.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable (in this case, bb) with a non-zero coefficient.

Q: How do I determine the degree of a polynomial difference?

A: To determine the degree of a polynomial difference, you need to analyze the resulting polynomial and identify the highest power of the variable with a non-zero coefficient.

Q: Can a polynomial difference be a binomial?

A: Yes, a polynomial difference can be a binomial if the resulting polynomial has exactly two terms.

Q: What is the difference between a polynomial and an algebraic expression?

A: A polynomial is a specific type of algebraic expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include a wide range of mathematical operations, including exponentiation, roots, and logarithms.

Q: How do I apply the distributive property when simplifying a polynomial difference?

A: When simplifying a polynomial difference, you need to apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial, and then combining like terms.

Q: Can a polynomial difference have a negative degree?

A: No, a polynomial difference cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

Q: How do I check my work when simplifying a polynomial difference?

A: To check your work when simplifying a polynomial difference, you need to verify that you have correctly applied the rules of polynomial arithmetic, including the distributive property, the commutative property, and the associative property.

Q: What are some common mistakes to avoid when simplifying polynomial differences?

A: Some common mistakes to avoid when simplifying polynomial differences include:

  • Failing to combine like terms
  • Applying the distributive property incorrectly
  • Ignoring the commutative and associative properties
  • Failing to check your work

Q: How do I use technology to simplify polynomial differences?

A: You can use technology, such as calculators or computer algebra systems, to simplify polynomial differences. However, it's essential to understand the underlying mathematical concepts and to verify the results using manual calculations.

Q: Can I use polynomial differences to solve real-world problems?

A: Yes, polynomial differences can be used to solve a wide range of real-world problems, including optimization problems, engineering design, and data analysis.