In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (3x⁴ y²+5x³ y−3y)−(2x⁴ y²−3x³ y−4y+6x)

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Start by distributing the negative sign across the second polynomial. This means you will change the sign of each term in the second polynomial: $(2x^4 y^2 - 3x^3 y - 4y + 6x)$ becomes $-2x^4 y^2 + 3x^3 y + 4y - 6x$.

Rewrite the expression by combining the first polynomial and the modified second polynomial: $(3x^4 y^2 + 5x^3 y - 3y) + (-2x^4 y^2 + 3x^3 y + 4y - 6x)$.

Group like terms together. Like terms are terms that have the same variables raised to the same powers. For example, group $3x^4 y^2$ with $-2x^4 y^2$, $5x^3 y$ with $3x^3 y$, $-3y$ with $4y$, and $-6x$ remains as is.

Combine the coefficients of the like terms. For example, $3x^4 y^2 - 2x^4 y^2$, $5x^3 y + 3x^3 y$, $-3y + 4y$, and $-6x$.

After simplifying, identify the degree of the resulting polynomial. The degree of a polynomial is determined by the term with the highest sum of the exponents of its variables. For example, in $x^4 y^2$, the degree is $4 + 2 = 6$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. In this context, we are focusing on subtraction, which involves combining like terms from two polynomials. Understanding how to identify and combine like terms is crucial for simplifying the expression correctly.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It provides insight into the polynomial's behavior and shape. After performing operations on polynomials, determining the degree helps classify the resulting polynomial and understand its characteristics.

Like Terms

Like terms are terms in a polynomial that have the same variable raised to the same power. For example, in the expression 3x^2 and 5x^2, both terms are like terms because they share the same variable and exponent. Recognizing and combining like terms is essential for simplifying polynomials and performing operations accurately.

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