Add or subtract, as indicated. $(3m^5-3m^2+4)+(-2m^3-m^2+6)$

Verified step by step guidance

1. Identify the polynomials to be added: \( (3m^5 - 3m^2 + 4) \) and \( (-2m^3 - m^2 + 6) \).
2. Remove the parentheses and write the expression as a single sum: \( 3m^5 - 3m^2 + 4 - 2m^3 - m^2 + 6 \).
3. Group like terms together. Like terms have the same variable raised to the same power: \( 3m^5 + (-2m^3) + (-3m^2 - m^2) + (4 + 6) \).
4. Combine the coefficients of the like terms by performing the indicated addition or subtraction: \( 3m^5 - 2m^3 - 4m^2 + 10 \).
5. Write the simplified polynomial as the final answer, ensuring terms are in descending order of exponents.

Key Concepts

Polynomial Addition and Subtraction

Adding or subtracting polynomials involves combining like terms, which are terms with the same variable raised to the same power. You add or subtract their coefficients while keeping the variable part unchanged. This process simplifies the expression into a single polynomial.

Like Terms

Like terms are terms in a polynomial that have identical variable parts, including the same exponents. For example, 3m^5 and 5m^5 are like terms, but 3m^5 and 2m^3 are not. Recognizing like terms is essential for correctly combining terms during addition or subtraction.

Polynomial Notation and Exponents

Polynomials are expressions consisting of variables raised to whole-number exponents and coefficients. Understanding the notation, such as m^5 meaning m raised to the fifth power, helps in identifying like terms and performing operations correctly.