Perform each division. See Examples 7 and 8. $(-4m^2{n}^2-21m{n}^3+18m{n}^2)/(-14m^2{n}^3)$

Verified step by step guidance

1. First, write the division expression clearly as a fraction: \(\frac{-4m^{2}n^{2} - 21mn^{3} + 18mn^{2}}{-14m^{2}n^{3}}\).
2. Next, separate the fraction into the sum of three fractions by dividing each term in the numerator by the denominator individually: \(\frac{-4m^{2}n^{2}}{-14m^{2}n^{3}} + \frac{-21mn^{3}}{-14m^{2}n^{3}} + \frac{18mn^{2}}{-14m^{2}n^{3}}\).
3. Simplify each fraction by canceling common factors in the numerator and denominator, such as powers of \(m\) and \(n\), and reduce coefficients by their greatest common divisor.
4. Rewrite each simplified fraction as a product of constants and variables with their exponents, making sure to handle negative signs carefully.
5. Finally, combine the simplified terms back into a single expression, if possible, to express the result of the division.

Key Concepts

Polynomial Division

Polynomial division involves dividing each term of the numerator by the denominator separately when the denominator is a monomial. This simplifies the expression by reducing powers and coefficients, making it easier to work with complex algebraic fractions.

Laws of Exponents

When dividing variables with exponents, subtract the exponent in the denominator from the exponent in the numerator for each variable. For example, m^a / m^b = m^(a-b). This rule helps simplify terms during division of algebraic expressions.

Simplifying Algebraic Fractions

Simplifying algebraic fractions means reducing the expression to its simplest form by canceling common factors in numerator and denominator. This often involves factoring terms and applying exponent rules to make the expression more manageable.