Polynomial Differences Determining The Degree And Type Of Resulting Expressions
When dealing with polynomials, understanding how to simplify and differentiate them is a fundamental skill in algebra. In this comprehensive guide, we will explore the process of finding the difference between two polynomials and then determining the characteristics of the resulting expression. Specifically, we will address the question: Which statement is true about the completely simplified difference of the polynomials a³b + 9a²b² - 4ab⁵ and a³b* - 3a²b² + ab*⁵? This question involves several key concepts, including polynomial subtraction, simplification by combining like terms, identifying the degree of a polynomial, and classifying polynomials by the number of terms. By the end of this article, you will have a solid understanding of these concepts and be able to confidently solve similar problems.
Defining Polynomials
Before we dive into the specifics of the problem, let's first define what polynomials are and review some important terminology. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials can have one or more terms, where each term is a product of a constant (coefficient) and variables raised to non-negative integer powers. For example, 3x² + 2x - 1 is a polynomial, while 2x⁻¹ + √x is not because it involves a negative exponent and a fractional exponent, respectively.
The degree of a term in a polynomial is the sum of the exponents of the variables in that term. For instance, in the term 5x³y², the degree is 3 + 2 = 5. The degree of a polynomial is the highest degree of any of its terms. For example, the degree of the polynomial 7x⁴ - 2x² + x is 4 because the term with the highest degree is 7x⁴.
Polynomials can be classified based on the number of terms they contain. A monomial has one term (e.g., 5x²), a binomial has two terms (e.g., 2x + 3), a trinomial has three terms (e.g., x² - 4x + 7), and so on. Polynomials with more than three terms are generally referred to as polynomials with that specific number of terms (e.g., a polynomial with four terms).
Subtracting Polynomials
To subtract one polynomial from another, we need to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x²y and -2x²y are like terms, while 4x² and 4x³ are not because the exponents of x are different.
The process of subtracting polynomials involves distributing the negative sign to each term in the polynomial being subtracted and then combining like terms. Let's consider two polynomials, P = 5x³ - 2x² + x - 4 and Q = 2x³ + x² - 3x + 1. To find P - Q, we first distribute the negative sign to each term in Q:
P - Q = (5x³ - 2x² + x - 4) - (2x³ + x² - 3x + 1)
= 5x³ - 2x² + x - 4 - 2x³ - x² + 3x - 1
Next, we combine like terms:
= (5x³ - 2x³) + (-2x² - x²) + (x + 3x) + (-4 - 1)
= 3x³ - 3x² + 4x - 5
So, the difference P - Q is 3x³ - 3x² + 4x - 5.
Solving the Specific Problem
Now, let's apply these concepts to the problem at hand. We are given two polynomials:
Polynomial 1: a³b + 9a²b² - 4ab*⁵
Polynomial 2: a³b - 3a²b² + ab*⁵
We need to find the difference between these polynomials and determine which statement about the difference is true.
First, let's subtract Polynomial 2 from Polynomial 1:
(a³b + 9a²b² - 4ab⁵) - (a³b* - 3a²b² + ab*⁵)
Distribute the negative sign:
= a³b + 9a²b² - 4ab⁵ - a³b* + 3a²b² - ab*⁵
Now, combine like terms:
= (a³b - a³b) + (9a²b² + 3a²b²) + (-4ab⁵ - ab⁵)
= 0a³b + 12a²b² - 5ab*⁵
= 12a²b² - 5ab*⁵
So, the simplified difference is 12a²b² - 5ab*⁵. Now, let's analyze the characteristics of this resulting polynomial.
Analyzing the Resulting Polynomial
The simplified difference, 12a²b² - 5ab⁵, has two terms: 12a²b² and -5ab⁵. Therefore, it is a binomial. To determine the degree of this binomial, we need to find the degree of each term.
The degree of the term 12a²b² is the sum of the exponents of a and b, which is 2 + 2 = 4.
The degree of the term -5ab*⁵ is the sum of the exponents of a and b, which is 1 + 5 = 6 (remember that if a variable has no visible exponent, it is understood to be 1).
The degree of the binomial is the highest degree of its terms, which is 6.
Evaluating the Given Statements
Now, let's evaluate the statements provided in the problem:
A. The difference is a binomial with a degree of 5.
B. The difference is a binomial with a degree of 6.
Based on our analysis, the difference is indeed a binomial, and its degree is 6. Therefore, statement B is true.
Conclusion
In summary, to find the simplified difference of the polynomials a³b + 9a²b² - 4ab⁵ and a³b* - 3a²b² + ab⁵, we subtracted the second polynomial from the first, combined like terms, and obtained 12a²b² - 5ab⁵. This resulting polynomial is a binomial with a degree of 6. Therefore, the correct statement is:
B. The difference is a binomial with a degree of 6.
This exercise highlights the importance of understanding polynomial operations, including subtraction and simplification. By mastering these concepts, you can confidently tackle more complex algebraic problems. Remember to always distribute the negative sign when subtracting polynomials and combine only like terms to achieve the correct simplified form.
Which of the following is true about the simplified difference between the polynomials a³b + 9a²b² - 4ab⁵ and a³b* - 3a²b² + ab*⁵?
