Problem 39 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{x^{2}+3 x}{x^{2}+x-12}-\frac{x^{2}-12}{x^{2}+x-12} $$

Step-by-Step Solution

Answer

The solution to the given algebraic fraction is \(\frac{15x}{(x-3)(x+4)}\).

Step 1: Identifying the denominators

Looking at the given expression, \(\frac{x^{2}+3 x}{x^{2}+x-12}-\frac{x^{2}-12}{x^{2}+x-12}\), both fractions possess the same denominators which is \(x^{2}+x-12\). This gives green light to perform the operation.

Step 2: Subtraction of the algebraic fractions

Once the denominators are confirmed to be the same, proceed with the subtraction operation by subtracting the numerators. That leads to \(\frac{(x^{2}+3x)-(x^{2}-12)}{x^{2}+x-12}\). This simplifies the numerator by getting rid of parentheses. It is worth noticing that the terms in the second parentheses will change signs due to the subtraction rule.

Step 3: Simplification of the resulting fraction

After simplification, the fraction will be \(\frac{15x}{x^{2}+x-12}\).

Step 4: Further simplification

The fraction can be further simplified by factoring the denominator. As a result, it will look like this \(\frac{15x}{(x-3)(x+4)}\)

Key Concepts

Subtraction of Fractions

Subtracting fractions in algebra involves ensuring the denominators are the same. When the denominators match, the numerators can be subtracted directly. This is similar to basic fraction subtraction learned in earlier math lessons, where the denominators have to be common.

For example, in the expression \( \frac{x^{2}+3x}{x^{2}+x-12}-\frac{x^{2}-12}{x^{2}+x-12} \), the denominators \( x^{2}+x-12 \) are already identical. This allows the subtraction to proceed smoothly by directly dealing with the numerators:\[ \frac{(x^{2}+3x)-(x^{2}-12)}{x^{2}+x-12} \]

Remember:

• Check for common denominators before subtracting.
• Change signs of the terms inside the second numerator when subtracting.
• Simplify the expression further as needed.

Factoring Polynomials

Factoring polynomials is a key skill in simplifying algebraic expressions. When dealing with fractions, factoring can make complex terms more manageable and is often necessary in the simplification process.

In our example, after subtracting the numerators, we simplify to \( \frac{15x}{x^{2}+x-12} \). The next step involves factoring the quadratic polynomial in the denominator:

• The expression \( x^{2}+x-12 \) can be factored into \( (x-3)(x+4) \).
• Factoring reveals the underlying structure and simplifies the fraction further.

Factoring often requires recognizing patterns or using the quadratic formula when necessary.

Understanding factoring will help with breaking down complex problems into easier parts.

Simplification of Expressions

Simplifying expressions involves both the numerators and denominators. Once you subtract the fractions and factor any polynomials, you look for opportunities to reduce the expression further.

With \( \frac{15x}{(x-3)(x+4)} \), you must ensure no further simplifications are possible. Check if any common factors exist between the numerator and the denominator.

Tips for Simplifying:

• Factor both the numerator and the denominator completely.
• Cancel out common factors if they are present.
• If no further factoring is possible, ensure the expression is in its simplest form.

This process not only makes the expression easier to understand but often reveals important properties or values integral to solving equations.