Problem 107 - Algebra and Trigonometry

Question

perform the indicated operations. $$ \left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right) $$

Step-by-Step Solution

Answer

The final simplified expression is \( \frac{x^2 - x + 2}{x(x+1)(x+2)(x+3)} \)

Step 1: Distribute Terms

Like terms are to be multiplied between brackets, following the distributive property. \( \left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right) = 1- \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)} \), similarly, perform the operation for \( \left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right) \)

Step 2: Combine Similar Terms

The next step is combining similar terms in both components found in Step 1. \( 1- \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{x(x+1) - x - (x+1) + 1}{x(x+1)} \) and similar simplification is applied to the second component.

Step 3: Simplify Numerators

Simplify numerators of both the components by adding and subtracting like terms.

Step 4: Combine Components

Finally, multiply the two components to get the final expression.

Key Concepts

The Distributive Property

The distributive property is a crucial tool in algebra that helps to simplify expressions. It allows you to multiply a single term by each term inside a parenthesis. In the expression \(\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\),we apply the distributive property by multiplying each term in the first parenthesis by each term in the second parenthesis.

• First, multiply \(1\) (the first term of the first parenthesis) by \(1\) and then by \(-\frac{1}{x+1}\).
• Next, multiply \(-\frac{1}{x}\) by \(1\) and then by \(-\frac{1}{x+1}\).

Simplifying these results,

1. \(1\times 1 = 1\)
2. \(1\times \left(-\frac{1}{x+1}\right) = -\frac{1}{x+1}\)
3. \(-\frac{1}{x}\times 1 = -\frac{1}{x}\)
4. \(-\frac{1}{x}\times\left(-\frac{1}{x+1}\right) = \frac{1}{x(x+1)}\)

This simplifies to:\[1 - \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)}\] Understanding this property makes it easier to tackle more complex expressions.

Combining Like Terms

Combining like terms is a fundamental concept in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression \(1 - \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)}\),we look for terms that can be combined to simplify the expression.

• In this case, \(1\) is a constant term.
• The terms \(-\frac{1}{x}\) and \(-\frac{1}{x+1}\) are fractional terms that must remain separate due to their different denominators.
• The term \(\frac{1}{x(x+1)}\) is unique and can't be combined with other terms.

So, the expression doesn't have like terms that can be combined directly, but organizing these properly in your simplification is crucial. Simplifying involves ensuring correct operation between terms and confirming if further simplification is possible.

Understanding Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. In the expression \(1 - \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)}\),we encounter rational expressions through the presence of fractions with polynomials in the denominator.

• A key step in simplifying them involves finding a common denominator if you’re adding or subtracting rational terms.
• In this context, for terms \(\frac{1}{x}\) and \(\frac{1}{x(x+1)}\), a common denominator must be found if they were to be combined.
• Multiplication of two rational expressions, like further operations in the original problem, involves multiplying numerators together and denominators together.

Conceptually, the idea is not much different from manipulating normal fractions. Remembering this can ease the perceived difficulty when working with rational expressions in algebra.