Problem 38 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{y^{3}+5 y^{2}}{y^{3}-16 y}-\frac{36 y}{y^{3}-16 y} $$

Step-by-Step Solution

Answer

\frac{y + 9}{y + 4}

Step 1: Factor the Denominator

First, factor the common denominator in each term. The common denominator in this case can be factored as follows: \[ y^3 - 16y = y(y^2 - 16) = y(y - 4)(y + 4) \]

Step 2: Rewrite Fractions with Common Denominator

Rewrite each fraction using the factored denominator: \[ \frac{y^3 + 5y^2}{y(y - 4)(y + 4)} - \frac{36y}{y(y - 4)(y + 4)} \]

Step 3: Combine the Fractions

Since the fractions have the same denominator, combine them into a single fraction: \[ \frac{(y^3 + 5y^2) - 36y}{y(y - 4)(y + 4)} \]

Step 4: Simplify the Numerator

Expand and then factor the numerator if possible: \[ y^3 + 5y^2 - 36y \] Group and factor by grouping: \[ y(y^2 + 5y - 36) \] Factor the trinomial: \[ y(y + 9)(y - 4) \]

Step 5: Cancel Common Factors

The term \(y(y + 9)(y - 4)\) in the numerator and \(y(y - 4)(y + 4)\) in the denominator have common factors that can be cancelled:\[ \frac{y(y + 9)(y - 4)}{y(y - 4)(y + 4)} \] The common factors \(y\) and \(y - 4\) are cancelled out, leaving: \[ \frac{y + 9}{y + 4} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a helpful technique when working with algebraic expressions. It involves breaking down a complicated polynomial into simpler products of polynomials. In this exercise, we started by factoring the common denominator. Consider the polynomial in the denominator, \(y^3 - 16y\). We can factor this as follows:

• Factor out a common factor of \( y \): \( y(y^2 - 16)\)
• Recognize that \( y^2 - 16 \) is a difference of squares: \((y - 4)(y + 4) \)
• Combine them into a complete factored form: \[ y(y - 4)(y + 4) \]

This expression is much simpler to work with. Understanding factoring will help you simplify complex expressions and make subsequent steps easier.

Common Denominators

When adding or subtracting fractions, it's essential to have a common denominator. This term enables the fractions to be written over the same denominator, allowing for straightforward combining.
In our exercise, both fractions already share the same factored denominator, \[ y(y - 4)(y + 4) \]. This lets us combine them as follows: \[ \frac{y^3 + 5y^2}{y(y - 4)(y + 4)} - \frac{36y}{y(y - 4)(y + 4)} = \frac{(y^3 + 5y^2) - 36y}{y(y - 4)(y + 4)} \] Having common denominators makes the process of adding or subtracting fractions straightforward as we only need to combine the numerators while keeping the denominator unchanged.

Fractions

Fractions are essential in algebra, as they represent division between two expressions. Simplifying fractions involves reducing them to their simplest form. In our exercise, we combined the fractions and then focused on simplifying the numerator. The steps were:

• Combine the numerators: \( y^3 + 5y^2 - 36y \)
• Factor the numerator: \[ y(y + 9)(y - 4) \]
• Cancel out common factors with the denominator: \[ \frac{y(y + 9)(y - 4)}{y(y - 4)(y + 4)} \]

After cancelling the common factors, we are left with a simplified fraction, \[ \frac{y + 9}{y + 4} \]. Reducing fractions in this manner simplifies the expression, making it easier to work with and understand overall.