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

Verified step by step guidance

1. First, rewrite the expression clearly by removing the parentheses, remembering to distribute the minus sign across the second group: \((6m^{4} - 3m^{2} + m) - (2m^{3} + 5m^{2} + 4m) + (m^{2} - m)\) becomes \$6m^{4} - 3m^{2} + m - 2m^{3} - 5m^{2} - 4m + m^{2} - m$.
2. Next, group like terms together. Like terms are terms that have the same variable raised to the same power. Grouping gives: \$6m^{4} + (-2m^{3}) + (-3m^{2} - 5m^{2} + m^{2}) + (m - 4m - m)$.
3. Now, combine the coefficients of each group of like terms by adding or subtracting them: For \(m^{4}\) terms: \$6m^{4}\(; for \)m^{3}\( terms: \)-2m^{3}\(; for \)m^{2}\( terms: \)-3 - 5 + 1\(; for \)m\( terms: \)1 - 4 - 1$.
4. Simplify each group by performing the arithmetic on the coefficients: Calculate the sum for \(m^{2}\) terms and for \(m\) terms separately.
5. Finally, write the simplified expression by combining all the simplified terms back together, maintaining the order from highest to lowest power.

Key Concepts

Polynomial Addition and Subtraction

This involves combining like terms from different polynomials by adding or subtracting their coefficients. When subtracting, distribute the negative sign across all terms in the polynomial being subtracted before combining.

Like Terms

Like terms are terms that have the same variable raised to the same power. Only like terms can be combined by adding or subtracting their coefficients to simplify the expression.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses. It is essential when subtracting polynomials to correctly apply the negative sign to each term inside the parentheses.