Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \frac{8 x^{2} y^{3}-10 x^{3} y}{2 x^{2} y} $$

Step-by-Step Solution

Answer

The polynomial is \(8y^2 - 5x\).

Step 1: Factor the Denominator

Identify the expression in the denominator, which is already set: \(2x^2y\).

Step 2: Separate the Terms in the Numerator

Break down the numerator into separate fractions. The expression in the numerator is \(8x^2y^3 - 10x^3y\). This can be written as two separate fractions: \(\frac{8x^2y^3}{2x^2y}\) and \(\frac{-10x^3y}{2x^2y}\).

Step 3: Simplify the First Fraction

Simplify \(\frac{8x^2y^3}{2x^2y}\):- The \(x^2\) and \(y\) terms in the numerator and denominator cancel out to leave \(8 \times \frac{y^3}{y} = 8y^2\).

Step 4: Simplify the Second Fraction

Simplify \(\frac{-10x^3y}{2x^2y}\):- Cancel \(x^2\) and \(y\) to get \(-5x\).

Step 5: Combine the Simplified Terms

Combine the results of the simplified fractions: \(8y^2 - 5x\). Thus, the polynomial expression is \(8y^2 - 5x\).

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator consist of polynomials. In their simplest form, they are quite similar to numerical fractions, like \(\frac{1}{2}\), but instead consist of variables and coefficients. For example, \(\frac{8x^2y^3}{2x^2y}\) is a rational expression. Understanding rational expressions is crucial because these can often appear convoluted due to the presence of variables. Simplifying them involves similar processes to numerical fractions:

• Identify and factor common terms in the numerator and the denominator.
• Cancel any common factors that appear both on the top and bottom.

By simplifying rational expressions, the complexity reduces, making them easier to understand and work with, as demonstrated by transforming \(\frac{8x^2y^3 - 10x^3y}{2x^2y}\) to \(8y^2 - 5x\). It re-emphasizes the power of simplification in algebraic manipulations.

Algebraic Fractions

Algebraic fractions represent ratios of algebraic expressions, where both the numerator and the denominator are polynomials. Each term in an algebraic fraction can be simplified separately by dividing out common factors. For instance, in the expression \(\frac{8x^2y^3}{2x^2y}\), both the numerator and the denominator have common variable factors.To simplify algebraic fractions, follow these steps:

• Factor out any common elements from both the numerator and the denominator.
• Simplify the fraction by canceling these common terms.
• Ensure that all variable powers are properly reduced.

Application of these steps means that \(x^2\) and \(y\) in both numerator and denominator "cancel out" to produce a simpler expression like \(8y^2\). Understanding simplification helps in clearer and more precise work with algebraic expressions.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential in simplifying expressions, solving equations, and understanding algebra better. Let's explore an example from exercise, the expression \(8x^2y^3 - 10x^3y\). Factoring involves:

• Identifying the greatest common factor (GCF) from all terms. Here, \(2x^2y\) is crucial as part of the denominator.
• Writing each term of the numerator by its essential factors and matching them to the GCF.
• Reconstructing the expression by dividing out the GCF from each term of the polynomial.

For example, \(8x^2y^3\) can be separated to \(8\cdot x^2\cdot y^3\), matching the \(x^2\) and \(y\) parts to the denominator, allowing us to cancel them properly. Factoring makes these simplifications and derivations much more manageable, underlining its importance as a fundamental tool in algebra.