Determining The Degree Of Sum And Difference Of Polynomials
Polynomials are fundamental building blocks in algebra, and understanding their properties is crucial for various mathematical applications. One essential property is the degree of a polynomial, which dictates its behavior and characteristics. When dealing with multiple polynomials, operations like addition and subtraction can lead to new polynomials with different degrees. This article aims to delve into the concept of the degree of the sum and difference of polynomials, specifically using the example of two given polynomials: 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³. We will explore how these operations affect the degree and provide a comprehensive analysis to clarify this concept.
Before diving into the specifics, let’s establish a clear understanding of polynomials and their degrees. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. For instance, 3x⁵y - 2x³y⁴ - 7xy³ is a polynomial. 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 3x⁵y, the degree is 5 (from x⁵) + 1 (from y), which equals 6. The degree of the polynomial itself is the highest degree among all its terms. Thus, understanding how to identify the degree of a polynomial is crucial for performing operations and analyzing their outcomes.
Degree of a Term
In a polynomial, each term is a product of constants and variables raised to non-negative integer powers. The degree of a term is the sum of the exponents of the variables in that term. For example:
- In the term 3x⁵y, the exponent of x is 5, and the exponent of y is 1. Therefore, the degree of this term is 5 + 1 = 6.
- In the term -2x³y⁴, the exponent of x is 3, and the exponent of y is 4. Thus, the degree of this term is 3 + 4 = 7.
- In the term -7xy³, the exponent of x is 1, and the exponent of y is 3. The degree of this term is 1 + 3 = 4.
Degree of a Polynomial
The degree of a polynomial is the highest degree of any of its terms. To find the degree of a polynomial, we first determine the degree of each term and then identify the highest among them. Considering the polynomial 3x⁵y - 2x³y⁴ - 7xy³:
- The degrees of the terms are 6, 7, and 4, respectively.
- The highest degree among these is 7.
Therefore, the degree of the polynomial 3x⁵y - 2x³y⁴ - 7xy³ is 7. This process is essential for classifying polynomials and understanding their behavior in algebraic operations.
When polynomials are added or subtracted, the resulting polynomial’s degree is determined by the highest degree terms that do not cancel each other out. To find the sum or difference, like terms (terms with the same variables raised to the same powers) are combined. The degree of the resulting polynomial is then identified. This operation is fundamental in simplifying complex algebraic expressions and solving equations.
Sum of Polynomials
To find the sum of two polynomials, we add like terms together. Consider the polynomials:
Polynomial 1: 3x⁵y - 2x³y⁴ - 7xy³
Polynomial 2: -8x⁵y + 2x³y⁴ + xy³
The sum is obtained as follows:
(3x⁵y - 2x³y⁴ - 7xy³) + (-8x⁵y + 2x³y⁴ + xy³)
= (3x⁵y - 8x⁵y) + (-2x³y⁴ + 2x³y⁴) + (-7xy³ + xy³)
= -5x⁵y + 0x³y⁴ - 6xy³
= -5x⁵y - 6xy³
In the resulting polynomial -5x⁵y - 6xy³, the terms have degrees 6 and 4, respectively. Therefore, the degree of the sum is 6. This example illustrates how adding polynomials involves combining like terms and identifying the highest degree in the resulting expression.
Difference of Polynomials
To find the difference of two polynomials, we subtract like terms. Using the same polynomials:
Polynomial 1: 3x⁵y - 2x³y⁴ - 7xy³
Polynomial 2: -8x⁵y + 2x³y⁴ + xy³
The difference is obtained as follows:
(3x⁵y - 2x³y⁴ - 7xy³) - (-8x⁵y + 2x³y⁴ + xy³)
= 3x⁵y - 2x³y⁴ - 7xy³ + 8x⁵y - 2x³y⁴ - xy³
= (3x⁵y + 8x⁵y) + (-2x³y⁴ - 2x³y⁴) + (-7xy³ - xy³)
= 11x⁵y - 4x³y⁴ - 8xy³
In the resulting polynomial 11x⁵y - 4x³y⁴ - 8xy³, the terms have degrees 6, 7, and 4, respectively. Therefore, the degree of the difference is 7. Understanding subtraction of polynomials is vital for solving algebraic problems where differences between expressions need to be determined.
Let's revisit the given polynomials and perform the operations to determine the degrees of the sum and difference. The polynomials are:
Polynomial 1: 3x⁵y - 2x³y⁴ - 7xy³
Polynomial 2: -8x⁵y + 2x³y⁴ + xy³
Finding the Sum
Adding the two polynomials:
(3x⁵y - 2x³y⁴ - 7xy³) + (-8x⁵y + 2x³y⁴ + xy³)
= 3x⁵y - 2x³y⁴ - 7xy³ - 8x⁵y + 2x³y⁴ + xy³
Combining like terms:
= (3x⁵y - 8x⁵y) + (-2x³y⁴ + 2x³y⁴) + (-7xy³ + xy³)
= -5x⁵y + 0x³y⁴ - 6xy³
= -5x⁵y - 6xy³
The resulting polynomial is -5x⁵y - 6xy³. The degrees of the terms are 6 and 4, respectively. Therefore, the degree of the sum is 6.
Finding the Difference
Subtracting the two polynomials:
(3x⁵y - 2x³y⁴ - 7xy³) - (-8x⁵y + 2x³y⁴ + xy³)
= 3x⁵y - 2x³y⁴ - 7xy³ + 8x⁵y - 2x³y⁴ - xy³
Combining like terms:
= (3x⁵y + 8x⁵y) + (-2x³y⁴ - 2x³y⁴) + (-7xy³ - xy³)
= 11x⁵y - 4x³y⁴ - 8xy³
The resulting polynomial is 11x⁵y - 4x³y⁴ - 8xy³. The degrees of the terms are 6, 7, and 4, respectively. Therefore, the degree of the difference is 7.
In conclusion, by performing the addition and subtraction of the polynomials 3x⁵y - 2x³y⁴ - 7xy³ and -8x⁵y + 2x³y⁴ + xy³, we found that the sum has a degree of 6, and the difference has a degree of 7. Therefore, the correct answer is:
The sum has a degree of 6, and the difference has a degree of 7.
This exercise underscores the importance of understanding polynomial degrees and how they change with algebraic operations. Understanding these concepts is essential for simplifying and solving more complex algebraic problems. Furthermore, this analysis highlights the significance of accurately performing polynomial arithmetic to determine the resulting degrees. Mastering these skills provides a solid foundation for advanced mathematical studies.
The sum has a degree of 6, and the difference has a degree of 7.
