Add or subtract, as indicated. See Example 4.(6m⁴ - 3m² + m) - (2m³ + 5m² + 4m) + (m² - m)

Verified step by step guidance

1. Step 1: Distribute the negative sign across the second polynomial: \(-(2m^3 + 5m^2 + 4m)\) becomes \(-2m^3 - 5m^2 - 4m\).
2. Step 2: Distribute the positive sign across the third polynomial: \((m^2 - m)\) remains \(m^2 - m\).
3. Step 3: Combine all the polynomials: \(6m^4 - 3m^2 + m - 2m^3 - 5m^2 - 4m + m^2 - m\).
4. Step 4: Group like terms together: \(6m^4\), \(-2m^3\), \(-3m^2 - 5m^2 + m^2\), \(m - 4m - m\).
5. Step 5: Simplify each group of like terms: Combine the coefficients of each group to simplify the expression.

Key Concepts

Polynomial Operations

Polynomial operations involve adding, subtracting, and multiplying polynomials, which are expressions consisting of variables raised to whole number powers. To add or subtract polynomials, like terms (terms with the same variable and exponent) must be combined. Understanding how to identify and group these like terms is essential for simplifying polynomial expressions.

Like Terms

Like terms are terms in a polynomial that have the same variable raised to the same power. For example, in the expression 3m² and -2m², both terms are like terms because they both contain the variable m raised to the power of 2. Recognizing and combining like terms is crucial for simplifying polynomial expressions accurately.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing for the multiplication of a single term by each term within a parenthesis. This property is often used in polynomial operations to ensure that all terms are accounted for when adding or subtracting polynomials. Mastery of this concept is vital for correctly simplifying expressions that involve parentheses.