Problem 58 - Algebra 2

Question

Which expression equals \(\frac{5 x}{x^{2}-9}-\frac{4 x}{x^{2}+5 x+6} ?\) A. \(\frac{7 x}{(x-3)(x+3)(x+2)}\) B. \(\frac{x^{2}-2 x}{(x-3)(x+3)(x+2)}\) C. \(\frac{x^{2}+22 x}{(x-3)(x+3)(x+2)}\) D. \(\frac{9 x^{2}-2 x}{(x-3)(x+3)(x+2)}\)

Step-by-Step Solution

Answer

The expression equals \(\frac{x^{2}-2 x}{(x-3)(x+3)(x+2)}\), so the correct answer is B.

Step 1: Factor denominators

Let's factor denominators of both fractions. The factored form of \(x^{2}-9\) is \((x-3)(x+3)\) and \(x^{2}+5x+6\) is \((x+2)(x+3)\).

Step 2: Get common denominator

In order to subtract fractions, they need to have the same denominator. The common denominator is the least common multiple (LCM) of denominators. So in this case, the LCM is \((x-3)(x+3)(x+2)\). Multiply the numerator and denominator of the first fraction by \((x+2)\) and the second fraction by \((x-3)\).

Step 3: Subtract the fractions

After getting a common denominator, subtract the numerators: \(\frac{5x(x+2)}{(x-3)(x+3)(x+2)} - \frac{4x(x-3)}{(x-3)(x+3)(x+2)}\). Simplify this to get \(\frac{x^2 - 2x}{(x-3)(x+3)(x+2)}\).

Key Concepts

Factoring Polynomials

To tackle the problem of subtracting rational expressions, we first need to factor polynomials in the denominator. Factoring polynomials involves expressing a polynomial as a product of its simplest factors. This is crucial because it helps simplify expressions and find common denominators.

For example, in the expression \(x^2 - 9\), we recognize it as a difference of squares. A difference of squares is factored as \((a-b)(a+b)\), where \(a=x\) and \(b=3\). Therefore, \(x^2 - 9\) factors into \((x-3)(x+3)\).

Similarly, \(x^2 + 5x + 6\) is a quadratic trinomial. To factor trinomials, we look for two numbers that multiply to the constant term (6) and add to the linear coefficient (5). These numbers are 2 and 3. So, \(x^2 + 5x + 6\) factors into \((x+2)(x+3)\).

Factoring makes it much easier to manipulate and combine these rational expressions.

Least Common Multiple (LCM)

When dealing with rational expressions, particularly in subtraction, finding the least common multiple (LCM) of the denominators is essential. This ensures that you have a common base to perform operations such as addition or subtraction.

To find the LCM of polynomial expressions like \((x-3)(x+3)\) and \((x+2)(x+3)\), you combine the factors from both expressions without repeating shared factors. The factor \((x+3)\) is common in both, so it is included only once. Therefore, the LCM is \((x-3)(x+3)(x+2)\).

This LCM serves as a universal denominator, allowing us to express both fractions in the exercise with equivalent denominators, making subtraction straightforward.

Simplifying Expressions

After obtaining a common denominator, the next step is to simplify the expression obtained by subtracting the numerators of the fractions. This involves straightforward algebraic manipulation.

In our example, we need to subtract the two expressions: \(\frac{5x(x+2)}{(x-3)(x+3)(x+2)} - \frac{4x(x-3)}{(x-3)(x+3)(x+2)}\). Begin by distributing the terms within each numerator:

• For the first term, distribute: \(5x(x+2) = 5x^2 + 10x\).
• For the second term, distribute: \(4x(x-3) = 4x^2 - 12x\).

Subtract the second result from the first:

• \(5x^2 + 10x - (4x^2 - 12x) = 5x^2 + 10x - 4x^2 + 12x\)
• Combine like terms: \(x^2 + 22x\).

The simplified expression after subtraction is thus \(\frac{x^2 + 22x}{(x-3)(x+3)(x+2)}\). This is simplified to the fullest extent given the scope of the exercise.