Problem 14 - Algebra 2

Question

Simplify. $$ \frac{5 x y^{2}-6 y^{3}+3 x^{2} y^{3}}{x y} $$

Step-by-Step Solution

Answer

The simplified expression is \(5y - \frac{6y^2}{x} + 3xy^2\).

Step 1: Break Down the Numerator

Identify each term in the numerator: the expression is \[ 5xy^2 - 6y^3 + 3x^2y^3 \] It consists of three terms: \(5xy^2\), \(-6y^3\), and \(3x^2y^3\).

Step 2: Separate the Fraction

Express each term in the numerator as individual fractions divided by the denominator \(xy\): \[ \frac{5xy^2}{xy} - \frac{6y^3}{xy} + \frac{3x^2y^3}{xy} \]

Step 3: Simplify Each Fraction

Simplify each fraction individually:1. \(\frac{5xy^2}{xy}\) simplifies to \(5y\) because the \(x\) and one \(y\) cancel out.2. \(\frac{6y^3}{xy}\) simplifies to \(\frac{6y^2}{x}\) because one \(y\) cancels out and \(x\) remains in the denominator.3. \(\frac{3x^2y^3}{xy}\) simplifies to \(3xy^2\) because one \(x\) and one \(y\) cancel out, leaving \(x\) and \(y^2\).

Step 4: Combine the Simplified Results

Combine the simplified fractions to get the final simplified expression:\[ 5y - \frac{6y^2}{x} + 3xy^2 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, consisting of numbers, variables, and operations. In the expression \( 5xy^2 - 6y^3 + 3x^2y^3 \), the terms are

• \( 5xy^2 \)
• \( -6y^3 \)
• \( 3x^2y^3 \)

These terms are separated by the plus or minus signs. Each term consists of:

• Coefficients, like 5, -6, and 3, which are the numerical parts of the terms
• Variables, such as \( x \) and \( y \), which can represent any number
• Exponents, which indicate how many times to use the variable in a multiplication

Understanding algebraic expressions is essential for simplifying more complex problems. Make sure you are familiar with identifying each part and understanding their roles in an expression.

Polynomial Simplification

Polynomial simplification involves reducing an expression to its most straightforward form without changing its value. This means we use operations such as addition and subtraction to combine like terms, and division for simplifying when fractions are involved.
In our example, we start by breaking down the numerator \( 5xy^2 - 6y^3 + 3x^2y^3 \) into separate fractional parts divided by \( xy \), then simplify each fraction individually.
Doing this results in:

• \( \frac{5xy^2}{xy} \) simplifies to \( 5y \) by canceling the common \( x \) and \( y \) terms.
• \( \frac{6y^3}{xy} \) reduces to \( \frac{6y^2}{x} \), since we cancel one \( y \) and the \( x \) remains as the denominator.
• \( \frac{3x^2y^3}{xy} \) simplifies to \( 3xy^2 \) by removing one \( x \) and \( y \) from the denominator and numerator.

Combine these simplified parts to achieve the final expression: \( 5y - \frac{6y^2}{x} + 3xy^2 \). This simplification helps in many mathematical scenarios, making it easier to evaluate, substitute values into, or further manipulate expressions.

Fractional Coefficients

Fractional coefficients arise when a term in a polynomial or expression is divided by another term, leaving a fraction as a coefficient. They are very common in algebra, especially during simplification processes.
In our solution:

• The term \( \frac{6y^2}{x} \) has a fractional coefficient because it retained an \( x \) in the denominator after simplification.

This signifies that any operations involving this term will need attention to both the numerator and denominator, such as finding common denominators for addition or subtraction.
Handling fractional coefficients generally involves ensuring that the expression is simplified completely, meaning that you may need to multiply all terms by the same factor to eliminate the fraction, only when necessary. In many cases, it is also about recognizing that the fraction affects the term's value like any integer coefficient would. When simplifying or solving expressions with fractions, be particularly careful to manage these coefficients properly to maintain accuracy in the solution.