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

Verified step by step guidance

1. Identify the division problem: \(\frac{15m^3 + 25m^2 + 30m}{5m^3}\).
2. Rewrite the division as separate fractions for each term in the numerator: \(\frac{15m^3}{5m^3} + \frac{25m^2}{5m^3} + \frac{30m}{5m^3}\).
3. Simplify each fraction by dividing the coefficients and subtracting the exponents of like bases: For example, \(\frac{15}{5} = 3\) and \(m^{3-3} = m^0 = 1\) for the first term.
4. Apply the same process to the other terms: \(\frac{25}{5} = 5\) and \(m^{2-3} = m^{-1}\); \(\frac{30}{5} = 6\) and \(m^{1-3} = m^{-2}\).
5. Write the simplified expression by combining the simplified terms: \(3 + 5m^{-1} + 6m^{-2}\).

Key Concepts

Polynomial Division

Polynomial division involves dividing each term of the numerator polynomial by the denominator, especially when the denominator is a monomial. This process simplifies the expression by reducing powers and coefficients accordingly.

Properties of Exponents

When dividing terms with the same base, subtract the exponents of the denominator from those of the numerator. For example, m^3 ÷ m^3 equals m^(3-3) = m^0 = 1, which helps simplify each term in the division.

Simplifying Rational Expressions

Simplifying rational expressions means reducing fractions by canceling common factors in numerator and denominator. After dividing each term, combine the simplified terms to write the expression in its simplest form.