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

Verified step by step guidance

1. Identify the expression to be divided: $-4m^2{n}^2-21m{n}^3+18m{n}^2$ and the divisor: $-14m^2{n}^3$.
2. Separate the division into individual terms: $(-4m^2{n}^2)/(-14m^2{n}^3)$, $(-21m{n}^3)/(-14m^2{n}^3)$, and $(18m{n}^2)/(-14m^2{n}^3)$.
3. Simplify each term by dividing the coefficients and subtracting the exponents of like variables.
4. For the first term: $(-4)/(-14)$ simplifies to $2/7$, and $m^2/m^2$ cancels out, while $n^2/n^3$ simplifies to $1/n$.
5. Repeat the simplification process for the remaining terms, ensuring to handle negative signs and exponents correctly.

Key Concepts

Polynomial Division

Polynomial division involves dividing a polynomial by another polynomial, similar to numerical long division. In this case, we divide the polynomial (-4m^2n^2 - 21mn^3 + 18mn^2) by the polynomial (-14m^2n^3). The process requires careful attention to the degrees of the terms and simplifying the resulting expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. In this problem, after performing the division, we will look for common factors between the terms of the numerator and the denominator to simplify the expression effectively.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. When simplifying expressions, it is important to recognize how negative exponents affect the terms, especially when dividing polynomials. Understanding this concept helps in rewriting and simplifying the final expression correctly.