Problem 53 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6} $$

Step-by-Step Solution

Answer

The simplified form of the provided expression is: \(\frac{x^2 - x}{(x - 2)(x + 3)(x + 5)}\)

Step 1: Factorization of Denominators

Begin by factoring the denominators of both fractions: \(x^{2} + 3x - 10\) factors out to \((x - 2)(x + 5)\) and \(x^{2} + x - 6\) factors out to \((x - 2)(x + 3)\)

Step 2: Identify the Least Common Denominator (LCD)

The LCD is identified by commoon factors in both the expressions. For denominators \((x - 2)(x + 5)\) and \((x - 2)(x + 3)\), the common factor is \((x - 2)\) and the unique factors are \((x + 5)\) and \((x + 3)\). So, the LCD would be \( (x - 2)(x + 5)(x + 3)\)

Step 3: Rewrite the expressions using the LCD

Rewriting the given fractions using the LCD: \(\frac{3x}{(x - 2)(x + 5)} - \frac{2x}{(x - 2)(x + 3)}\), the first fraction is missing \((x + 3)\) from its denominator and the second fraction is missing \((x + 5)\) from its denominator. Hence, multiply both the numerator and denominator of first fraction by \((x + 3)\) and the second fraction by \((x + 5)\)

Step 4: Perform Subtraction

After adjustment, the equation becomes: \(\frac{3x(x + 3)}{(x - 2)(x + 3)(x + 5)} - \frac{2x(x + 5)}{(x - 2)(x + 5)(x + 3)}\). Now, the common denominators allow us to easily subtract the numerators: \(\frac{3x^2 + 9x - 2x^2 - 10x}{ (x - 2)(x + 3)(x + 5)}\)

Step 5: Simplify the Result

Simplify the equation by combining like terms in the numerator: \(\frac{x^2 - x}{(x - 2)(x + 3)(x + 5)}\) which can't be simplified any further.

Key Concepts

Factoring Polynomials

When dealing with rational expressions, factoring polynomials is a critical skill. In our exercise, both denominators are quadratic polynomials: \(x^2 + 3x - 10\) and \(x^2 + x - 6\). Polynomials like these often require factoring into simpler binomials. This is done to break down expressions into their simplest components, making it easier to manipulate the fractions later.

For example:

• To factor \(x^2 + 3x - 10\), find two numbers that multiply to -10 and add to 3, which are -2 and 5. Thus, it factors to \((x - 2)(x + 5)\).
• Similarly, factor \(x^2 + x - 6\) into \((x - 2)(x + 3)\) by finding numbers that multiply to -6 and add to 1, which are -2 and 3.

This step sets the foundation for finding common denominators and simplifying expressions.

Least Common Denominator

When adding or subtracting rational expressions, finding a common denominator is crucial. This common ground allows the expressions to be combined or compared. Here, the denominators are \(x - 2)(x + 5)\) and \(x - 2)(x + 3)\). Although they share a common factor of \(x - 2\), they differ otherwise.

To find the Least Common Denominator (LCD), include all distinct factors. The common factor \(x - 2\) appears only once, while the other factors \(x + 5\) and \(x + 3\) are included to ensure each original denominator is represented fully. Thus, the LCD is:

• \((x - 2)(x + 5)(x + 3)\)

Once we have the LCD, we can rewrite fractions with this denominator, facilitating the addition or subtraction process.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. In our task, after rewriting the fractions using the LCD, the subtraction can proceed, leading us to simplify by combining terms.

The equation becomes:

• \(\frac{3x(x + 3)}{(x - 2)(x + 3)(x + 5)} - \frac{2x(x + 5)}{(x - 2)(x + 3)(x + 5)}\)

Notice how we can now focus on the numerators since the denominators are identical.

Combine like terms in the numerators to achieve the simplest form:

• \(3x(x+3)\) expands to \(3x^2 + 9x\)
• \(2x(x+5)\) expands to \(2x^2 + 10x\)
• Subtracting the second from the first gives \(3x^2 + 9x - 2x^2 - 10x = x^2 - x\)

This leaves the final fraction as \(\frac{x^2 - x}{(x - 2)(x + 3)(x + 5)}\), which is already in its simplest form.

Algebraic Subtraction

Subtraction in algebra works similarly to basic arithmetic subtraction but with a focus on equivalent expressions. When dealing with rational expressions, once you've aligned the denominators, subtract the numerators and retain the common denominator.

In our example, after rewriting with the LCD, the expressions in the numerator were:

• Numerator of the first term: \(3x(x + 3)\)
• Numerator of the second term: \(2x(x + 5)\)

With a common denominator of \((x - 2)(x + 3)(x + 5)\), subtraction becomes straightforward:

Simply perform \(3x^2 + 9x - (2x^2 + 10x)\), which leaves \(x^2 - x\).

This clear algebraic method ensures accuracy and allows for easy verification of your work.