Perform each division. See Examples 7 and 8. (8wxy^2+3wx^2y+12w^2xy)/(4wx^2y)

Verified step by step guidance

1. Identify the common factor in the numerator and the denominator. The denominator is $4w{x}^{2}y$.
2. Break down each term in the numerator $8wx{y}^2+3w{x}^{2}y+12w^{2}xy$ and divide by the denominator $4w{x}^{2}y$.
3. For the first term $\frac{8wx{y}^2}{4w{x}^{2}y}$, simplify by canceling common factors: $\frac{8}{4}=2$, $w$ cancels out, $y$ cancels out, and $x$ in the denominator reduces the power of $x$ in the numerator.
4. For the second term $\frac{3w{x}^{2}y}{4w{x}^{2}y}$, simplify by canceling all common factors: $\frac{3}{4}$ remains, and all variables cancel out.
5. For the third term $\frac{12w^{2}xy}{4w{x}^{2}y}$, simplify by canceling common factors: $\frac{12}{4}=3$, $w$ in the denominator reduces the power of $w$ in the numerator, $x$ cancels out, and $y$ cancels out.

Key Concepts

Polynomial Division

Polynomial division involves dividing a polynomial by another polynomial or a monomial. In this case, we are dividing the polynomial expression in the numerator by the monomial in the denominator. The process requires simplifying each term of the polynomial by the term in the denominator, which can help in reducing the expression to its simplest form.

Factoring Out Common Terms

Factoring out common terms is a crucial step in simplifying algebraic expressions. When dividing polynomials, identifying and factoring out common variables and coefficients can make the division process easier. This involves recognizing shared factors in the numerator and denominator, allowing for cancellation and simplification of the expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling out common factors. This is essential in algebra to make calculations easier and clearer. In the context of the given problem, after performing the division, the resulting expression should be simplified by eliminating any common factors between the numerator and the denominator.