Problem 12 - Algebra A Combined Function

Question

$$ \frac{3 y}{y^{2}+3 y-10}-\frac{6}{y^{2}+3 y-10} $$

Step-by-Step Solution

Answer

\( \frac{3}{y + 5} \)

Step 1: Identify Common Denominator

Both terms in the expression share a common denominator, \( y^2 + 3y - 10 \). This common denominator simplifies the subtraction process.

Step 2: Combine the Numerators

Since the denominators are the same, subtract the numerators directly: \( \frac{3y - 6}{y^2 + 3y - 10} \).

Step 3: Factor the Numerator

Factor the numerator, \( 3y - 6 \), by taking out the greatest common factor, 3: \( 3(y - 2) \).

Step 4: Write Simplified Fraction

The expression simplifies to: \( \frac{3(y - 2)}{y^2 + 3y - 10} \).

Step 5: Factor the Denominator

Factor the quadratic expression in the denominator: \( y^2 + 3y - 10 = (y + 5)(y - 2) \).

Step 6: Simplify the Expression

Observe that \( y - 2 \) is a common factor in both the numerator and the denominator. Cancel it out: \( \frac{3}{y + 5} \).

Key Concepts

Factoring Polynomials

Polynomials often look complex, but factoring them can break them down into simpler components. This process essentially means rewriting a polynomial as a product of its factors.
To factor a polynomial, you might look for:

• Common factors, which are numbers or variables common to every term.
• Special patterns, like a difference of squares
• Recognizable quadratic forms that can be split into binomials.

An example in the solution provided is the polynomial in the denominator, \(y^2 + 3y - 10\). It was factored into two binomials, \((y + 5)(y - 2)\), using techniques associated with quadratic equations. Understanding how to factor polynomials correctly is essential for simplifying rational expressions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means converting a fraction so that the numerator and denominator share no common factors beyond 1.
In the exercise, we simplified the fraction \(\frac{3(y - 2)}{y^2 + 3y - 10}\).
The critical point is identifying and canceling common factors present in both the numerator and the denominator.
For instance, \((y - 2)\) is a factor that appears in both the numerator and the denominator, so it can be canceled out. This reduces the fraction to \(\frac{3}{y + 5}\).
Remember that each step in simplifying fractions makes calculations easier and neater.

Common Denominators

A common denominator in fraction arithmetic makes operations straightforward.
When working with rational expressions, having a common denominator allows you to directly add, subtract, or compare fractions.
In our problem, both fractions shared the denominator \(y^2 + 3y - 10\). This commonality allowed for straightforward subtraction of the numerators, leading to a single, simplified expression.
Using a common denominator helps maintain consistency and simplicity across your calculations.