Perform each division. See Examples 7 and 8. (15m^3+25m^2+30m)/(5m^3)

Verified step by step guidance

1. Identify the terms in the numerator: \(15m^3 + 25m^2 + 30m\).
2. Identify the term in the denominator: \(5m^3\).
3. Divide each term in the numerator by the term in the denominator separately: \(\frac{15m^3}{5m^3}\), \(\frac{25m^2}{5m^3}\), \(\frac{30m}{5m^3}\).
4. Simplify each division: \(\frac{15}{5}m^{3-3}\), \(\frac{25}{5}m^{2-3}\), \(\frac{30}{5}m^{1-3}\).
5. Combine the simplified terms to express the final result.

Key Concepts

Polynomial Division

Polynomial division is the process of dividing one polynomial by another, similar to numerical long division. In this case, we divide the polynomial in the numerator, 15m^3 + 25m^2 + 30m, by the monomial in the denominator, 5m^3. The result is obtained by dividing each term of the numerator by the term in the denominator, simplifying the expression step by step.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form. This includes combining like terms and factoring when possible. In the context of polynomial division, simplifying the result after division helps in understanding the behavior of the polynomial and can reveal important characteristics such as roots or intercepts.

Coefficients and Exponents

Coefficients are the numerical factors in a term of a polynomial, while exponents indicate the power to which the variable is raised. Understanding how to manipulate coefficients and exponents is crucial in polynomial division, as it affects the outcome of each division step. For example, when dividing terms like 15m^3 by 5m^3, both the coefficients and the exponents are divided, leading to a simplified result.