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

Verified step by step guidance

1. Identify the polynomials involved in the operation: \((3x^4 y^2 + 5x^3 y - 3y)\) and \((2x^4 y^2 - 3x^3 y - 4y + 6x)\).
2. Distribute the negative sign across the second polynomial: \(-(2x^4 y^2 - 3x^3 y - 4y + 6x) = -2x^4 y^2 + 3x^3 y + 4y - 6x\).
3. Combine like terms from both polynomials: \((3x^4 y^2 - 2x^4 y^2) + (5x^3 y + 3x^3 y) + (-3y + 4y) - 6x\).
4. Simplify each group of like terms: \(x^4 y^2\) terms, \(x^3 y\) terms, \(y\) terms, and the \(x\) term.
5. Determine the degree of the resulting polynomial by identifying the term with the highest sum of exponents.

Key Concepts

Polynomial Operations

Polynomial operations involve addition, subtraction, and multiplication of polynomial expressions. When performing these operations, like terms (terms with the same variable raised to the same power) are combined, while unlike terms remain separate. Understanding how to manipulate these expressions is crucial for simplifying and solving polynomial equations.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It provides insight into the polynomial's behavior and characteristics, such as the number of roots and the end behavior of its graph. When combining polynomials, the degree of the resulting polynomial is determined by the term with the highest degree after simplification.

Like and Unlike Terms

Like terms are terms that have the same variable raised to the same power, allowing them to be combined through addition or subtraction. Unlike terms, on the other hand, cannot be combined in this way. Recognizing and correctly identifying like and unlike terms is essential for accurately performing polynomial operations and simplifying expressions.