Problem 26 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{5 x^{3} y^{3}-15 x^{2} y^{2}+20 x y}{-5 x y} $$

Step-by-Step Solution

Answer

Answer: The quotient is $$-x^2y^2 + 3xy - 4$$.

Step 1: Identify the terms in the polynomial and the monomial

First, let's identify the terms in the polynomial and the monomial. The polynomial has three terms: $$5x^3y^3$$, $$-15x^2y^2$$, and $$+20xy$$. The monomial is $$-5xy$$.

Step 2: Divide each term of the polynomial by the monomial

Now, we will divide each term of the polynomial by the monomial $$(-5xy)$$: 1. $$\frac{5x^3y^3}{-5xy}$$ 2. $$\frac{-15x^2y^2}{-5xy}$$ 3. $$\frac{20xy}{-5xy}$$

Step 3: Simplify each fraction

Next, we will simplify each fraction: 1. $$\frac{5x^3y^3}{-5xy} = -x^2y^2$$ (divide coefficients and subtract exponents) 2. $$\frac{-15x^2y^2}{-5xy} = 3xy$$ (divide coefficients and subtract exponents) 3. $$\frac{20xy}{-5xy} = -4$$ (divide coefficients and subtract exponents)

Step 4: Combine the simplified fractions into the quotient

Finally, combine the simplified fractions from Step 3 into the final quotient: $$-x^2y^2 + 3xy - 4$$

Key Concepts

Monomial Division

Monomial division involves dividing a polynomial by a single term, which is known as a monomial. In the context of this exercise, the polynomial is divided by the monomial \(-5xy\). The key to division with a monomial is dividing each term of the polynomial individually by the monomial. This process is straightforward and ensures a clear pathway to the solution.

• Identify Terms: Spot the terms in both the polynomial and the monomial. Here, it is important to note that the polynomial consists of terms such as \(5x^3y^3\), while the monomial is \(-5xy\).
• Divide Each Term: Work through each term of the polynomial. Divide by the monomial separately, without rushing or combining steps, as exemplified by dividing \(\frac{5x^3y^3}{-5xy}\).
• Simple Steps: Simplifying intermediate results after each division step can make managing larger algebraic expressions easier.

This approach makes polynomial division more manageable, especially when working with numerous terms.

Algebraic Expressions

Algebraic expressions like polynomials are at the core of algebra, consisting of variables, coefficients, and exponents. Understanding them is key to performing operations such as division. In this specific exercise, the expression \(5x^3y^3 - 15x^2y^2 + 20xy\) is a classic polynomial.

• Structures: Each term of an algebraic expression is a product of a number and one or more variables raised to a power.
• Operations: Operations include addition, subtraction, and multiplication of these terms, often involving multiple variables as seen in the provided polynomial.
• Sign Importance: Be attentive to the signs in front of each term. It’s crucial during division that negative signs are maintained correctly.

Navigating algebraic structures and manipulating them according to the rules of arithmetic sets a strong foundation for mastering more complex algebraic duties.

Exponent Rules

Exponent rules are pivotal when dividing polynomials by a monomial, as they dictate how terms are simplified. The fundamental principle involves subtracting the exponent of the denominator from the exponent of the numerator for each term.

• Division Rule: When dividing terms with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\). This simplifies expressions such as \(x^3 / x = x^{3-1} = x^2\).
• Negative Exponents: Keep an eye out for negative exponents which indicate reciprocal relations, but in this situation, focus on subtraction of exponents.
• Simplifying Fractions: Alongside division of exponents, ensure coefficients are simplified normally (e.g., \(\frac{-15}{-5} = 3\)).

By using exponent rules appropriately, algebraic expressions can be divided cleanly, leading to simpler and more understandable results.