Perform each division. See Examples 7 and 8. $(8wx{y}^2+3w{x}^{2}y+12w^{2}xy)/\left(4w{x}^{2}y\right)$

Verified step by step guidance

1. Identify the division problem: \(\frac{8wxy^{2} + 3wx^{2}y + 12w^{2}xy}{4wx^{2}y}\).
2. Rewrite the expression as the sum of three separate fractions: \(\frac{8wxy^{2}}{4wx^{2}y} + \frac{3wx^{2}y}{4wx^{2}y} + \frac{12w^{2}xy}{4wx^{2}y}\).
3. Simplify each fraction individually by canceling common factors in the numerator and denominator:
4. - For \(\frac{8wxy^{2}}{4wx^{2}y}\), cancel common variables and coefficients.
5. - For \(\frac{3wx^{2}y}{4wx^{2}y}\), cancel common variables and coefficients.
6. - For \(\frac{12w^{2}xy}{4wx^{2}y}\), cancel common variables and coefficients.
7. After simplifying each term, combine the results to write the final simplified expression.

Key Concepts

Polynomial Division

Polynomial division involves dividing one polynomial by another, similar to numerical division. Each term in the numerator is divided by the denominator, simplifying the expression step-by-step. Understanding how to divide terms with variables and exponents is essential.

Laws of Exponents

The laws of exponents govern how to handle powers when multiplying, dividing, or raising powers to powers. For division, subtract the exponents of like bases. This is crucial when simplifying terms like w^a / w^b or x^a / x^b in the expression.

Combining Like Terms

After dividing each term, combining like terms means adding or subtracting terms with the same variables raised to the same powers. This step simplifies the expression to its simplest form, making it easier to interpret or use in further calculations.