Problem 63 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{6 x^{4} y^{4}+30 x^{4} y^{3}-x^{2} y^{2}+3 x y}{6 x^{2} y^{2}}$$

Step-by-Step Solution

Answer

The simplified expression after division is \(xy^2+5xy-x+\frac{3}{2}\).

Step 1: Factorize the Numerator

First, let's find the common factors of all terms in the given expression. We will factor out \(6x^2y^2\), that is the highest common factor among all the terms: \[\frac{6x^4y^4+30x^4y^3-x^2y^2+3xy}{6x^2y^2} = \frac{6x^2y^2(xy^2+5xy-x+3\frac{1}{2})}{6x^2y^2}\]

Step 2: Division by the Common Factor

Now that we have factored out the common factor, we can divide the entire expression by the common factor \(6x^2y^2\): \[\frac{6x^2y^2(xy^2+5xy-x+\frac{3}{2})}{6x^2y^2} = xy^2+5xy-x+\frac{3}{2}\] So the simplified expression is: \[xy^2+5xy-x+\frac{3}{2}\]

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial expression into simpler factors that, when multiplied together, give back the original expression. In the exercise provided, the goal is to factor out common terms from the polynomial in the numerator. Factoring is helpful because it allows us to simplify expressions by canceling out terms.

• Identify Terms: Look for terms that appear in each part of the expression. In our example, the common term in the numerator is \(6x^2y^2\).
• Extract Common Factor: By factoring \(6x^2y^2\) out of every term, the expression inside the parentheses becomes much simpler.
• Result: After factoring, we have \(6x^2y^2(xy^2 + 5xy - x + \frac{3}{2})\), making it easier to divide and reduce the fraction in the subsequent steps.

Understanding factoring aids in simplifying rational expressions, making calculations less cumbersome.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To handle these expressions effectively, it’s crucial to understand the concept of simplification through factoring and division. In our example, the expression \(\frac{6x^4y^4 + 30x^4y^3 - x^2y^2 + 3xy}{6x^2y^2}\) is a rational expression.

• Simplify the Numerator: By factoring out common terms from the numerator first, we make the whole expression easier to understand and manage.
• Dividing the Whole Expression: Once we have factored out the greatest common factor, simplifying the rational expression becomes straightforward, allowing us to cancel out identical terms from numerator and denominator.
• Result: The division results in \(xy^2 + 5xy - x + \frac{3}{2}\), a much simpler expression than the original.

Managing rational expressions through factoring and simplifying is an essential skill in algebra.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process makes expressions easier to understand and use in further calculations. In the exercise, simplification was achieved by factoring the numerator, and then dividing by a common factor.

• Streamline the Calculation by simplifying, especially when dealing with lengthy polynomial numerators.
• Canceling Common Terms: After factoring, we see that \(6x^2y^2\) can be canceled from both the numerator and the denominator.
• Final Result: By simplifying the expression \(xy^2 + 5xy - x + \frac{3}{2}\), it becomes manageable and ready for further operations or problem-solving tasks.

Simplifying expressions often makes solving equations or understanding relationships between variables much clearer.