Problem 10 - Algebra A Combined Function

Question

$$ \frac{x^{2}+9 x}{x+7}-\frac{4 x+14}{x+7} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^2 + 5x - 14}{x + 7} \).

Step 1: Identify the Common Denominator

Both fractions have the common denominator \(x + 7\). This allows us to combine the fractions into a single fraction.

Step 2: Combine the Numerators

Since the denominators are the same, we combine the numerators: \(x^2 + 9x - (4x + 14)\).

Step 3: Simplify the Numerator

Distribute the negative sign in \(-(4x + 14)\) and combine like terms:\[x^2 + 9x - 4x - 14 = x^2 + 5x - 14\]

Step 4: Simplify the Fraction

Rewrite the expression as a single fraction:\[ \frac{x^2 + 5x - 14}{x + 7} \]

Step 5: Factor the Numerator (optional if not simplifiable further)

You can try factoring the numerator \(x^2 + 5x - 14\) to see if further simplification is possible, however in this case it does not simplify neatly. Therefore, the final expression is kept as it is.

Key Concepts

Common Denominator

When dealing with algebraic fractions, finding a common denominator is essential for combining fractions into a single, simpler fraction. In this exercise, both fractions shared the denominator \(x + 7\). A common denominator means each fraction has the same lower part.

This helps us in two significant ways:

• Combines different algebraic fractions into one single fraction by aligning them over the same denominator.
• Keeps the fractions organized, reducing the complexity of multiple expressions.

By ensuring the denominators are the same, we focused attention entirely on smoothing out the numerators, knowing they'll neatly sit over a unified denominator. In this problem, using \(x + 7\) as the common denominator allowed us to proceed with combining the numerators seamlessly.

Numerator Simplification

Simplifying the numerator is a key skill once we've aligned the denominators. In our exercise, the original numerators were \(x^2 + 9x\) and \(4x + 14\). We combine these into one expression: \(x^2 + 9x - (4x + 14)\).

Here are the steps for simplification:

• Distribute the negative across the second fraction's numerator, changing signs: \(-(4x + 14)\) becomes \(-4x - 14\).
• Combine like terms: Add or subtract all terms that match in degree, such as \(x\) terms and constant terms.

So, the combined terms \(x^2 + 9x - 4x - 14\) become \(x^2 + 5x - 14\). This results in a tidy, single numerator over our common denominator \((x + 7)\). Every simplification makes the problem clearern by expressing its simplest form without altering the meaning.

Factoring Polynomials

When simplifying fractions, factoring polynomials can often further reduce the expression. Unfortunately, not every polynomial factors nicely, as is the case here with \(x^2 + 5x - 14\).

However, it's important to understand the concept:

• Look for two numbers that multiply to give the constant term (\(-14\)) and add to the middle coefficient (\(5\)).
• Test each possible factor to see if they balance both needs.

While some polynomials easily factor this way, others may defy simple factoring rules, leaving the polynomial unchanged. For this problem, as attempts to factor didn’t lead to a cleaner answer, we retain \(\frac{x^2 + 5x - 14}{x + 7}\). Remember, whether further simplification occurs or not, this process is vital to mastering polynomial operations in algebra.