Problem 76 - Algebra and Trigonometry

Question

perform the indicated operations. Simplify the result, if possible. $$ \left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{4x^2+25x+28}{x^2+x-2}\)

Step 1: Distribute the Terms

Multiply each term from the first parenthesis by each term from the second parenthesis like follows: \[ 4 * 1 + 4 * \frac{5}{{x-1}} - \frac{3}{{x+2}} * 1 - \frac{3}{{x+2}} * \frac{5}{{x-1}} \]

Step 2: Simplify the expression

Write down the multiplication results: \[ 4 + \frac{20}{{x-1}} - \frac{3}{{x+2}} - \frac{15}{{(x+2)*(x-1)}} \] Now we need to find a common denominator for these terms to simplify the expression. A good choice is \((x-1)(x+2)\) which is the least common multiple of the denominators.

Step 3: Simplify further

Rewrite the expression with common denominator: \[ \frac{4(x-1)(x+2) + 20(x+2) - 3(x-1) - 15}{(x-1)(x+2)} \]. After performing all the operations, we get: \[ = \frac{4x^2+8x + 20x + 40 - 3x + 3 - 15}{x^2+x-2} \]. Further simplifying brings to: \[ = \frac{4x^2+25x+28}{x^2+x-2} \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental principle in algebra that helps simplify complex problems. It allows you to break down problems involving multiplication over addition or subtraction. Simply put, it lets you distribute a single term across the terms inside a parenthesis.
For the given problem, we apply the distributive property to multiply each term in one set of parentheses by each term in the other set. This is done as follows:

• Multiply 4 by 1 resulting in 4.
• Multiply 4 by \( \frac{5}{x-1} \) resulting in \( \frac{20}{x-1} \).
• Multiply \( -\frac{3}{x+2} \) by 1 resulting in \( -\frac{3}{x+2} \).
• Multiply \( -\frac{3}{x+2} \) by \( \frac{5}{x-1} \) resulting in \( -\frac{15}{(x+2)(x-1)} \).

Each of these steps follows the distributive rule, allowing us to further combine and simplify the expression later. It is essential for solving the equation accurately.

Rational Expressions

Rational expressions are fractions that involve polynomials in the numerator, the denominator, or both. Dealing with rational expressions requires careful attention to the factors in both the numerator and the denominator.
In this exercise, after applying the distributive property, we are left with the expression:

• \( 4 + \frac{20}{x-1} - \frac{3}{x+2} - \frac{15}{(x+2)(x-1)} \)

To deal with these rational expressions, we need to find a common denominator. This allows us to combine and simplify the terms. The common denominator here is \((x-1)(x+2)\), which is formed by multiplying the different denominators together. This ensures that each term is expressed over the same base, making it easier to simplify.

Simplifying Expressions

Simplifying expressions is the process of reducing them to a more manageable form. After finding a common denominator for the rational expressions, we can then combine and simplify the equation.
The steps include:

• Multiply each term by the common denominator \((x-1)(x+2)\) to eliminate fraction parts.
• Combine like terms. This involves combining all similar terms from the expanded form.

We end up with:

• Expanded numerator: \[ 4(x-1)(x+2) + 20(x+2) - 3(x-1) - 15 \]

Simplifying gives us the final expression:

• \( \frac{4x^2 + 25x + 28}{x^2 + x - 2} \)

By following these steps, you can ensure that your solution is both clear and complete. Simplifying makes the expression much easier to handle, especially when tackling more complex algebraic problems.