Problem 4 - Elementary and Intermediate Algebra

Question

Simplify each expression. \(\frac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{(x+1)(x-9)} \).

Step 1: Identify common factors in the numerator and denominator

In the expression \( \frac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)} \), notice that there are common terms in the numerator and the denominator. Both the numerator and the denominator contain the factor \((x+7)\).

Step 2: Cancel common factors

Since \((x+7)\) is present in both the numerator and the denominator, we can cancel \((x+7)\) from both, which simplifies the expression to \( \frac{2 \cdot 5}{5(x+1)(x-9)} \).

Step 3: Simplify the remaining terms

Now, we notice that both the numerator and the denominator have a factor of 5. Cancel the 5 in the numerator and denominator, resulting in the expression \( \frac{2}{(x+1)(x-9)} \).

Step 4: Simplified expression

The simplified expression is now \( \frac{2}{(x+1)(x-9)} \). There are no more common factors that can be canceled, and the expression is fully simplified.

Key Concepts

Factoring

Factoring is a key skill in algebra that involves breaking down expressions into components, known as factors, that when multiplied together give you the original expression. In many algebra problems, identifying factorable parts is the first step towards simplification. Let's look at the expression \[ rac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)} \]

• Observe the terms: In the numerator, you have factors of \((x+7) \) and numbers 2 and 5. In the denominator, there’s a factor \((x+7)(x+1)(x-9)\)\ plus 5.
• Factoring allows you to see common terms which might cancel later. Here, \((x+7) \) is a prominent common factor.
• The goal of factoring in this context is to simplify the expression by identifying parts that will cancel each other out in subsequent steps, leading to a simpler form.

Breaking down expressions into simpler components through factoring not only helps in simplifying expressions but also makes solving equations much easier.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions involves similar processes as simplifying numerical fractions. Let's consider an example:\[ rac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)} \]

• A rational expression is fully simplified when it has no common factors left in the numerator and the denominator.
• This involves identifying parts of the expression that are identical in both the numerator and the denominator. For example, the factor \((x+7)\) appears in both.
• Once similar terms are spotted, we can proceed to cancel them, ensuring to check for any restrictions such as \((x eq 9) \) and \((x eq -1)\) that would make the original denominator zero.

Simplifying rational expressions is fundamental in algebra and calculus, helping reduce complexity and making equations more straightforward to interpret and solve.

Canceling Terms

Canceling terms is an essential simplification process. It reduces complex expressions into simpler forms by eliminating identical factors from both the numerator and the denominator of a fraction. Consider our example:\[ rac{(x+7) \cdot 2 \cdot 5}{5(x+1)(x+7)(x-9)} \]

• The first step is canceling \(x+7\), a common factor in both the numerator and denominator.
• Next, the number 5 is canceled since it is a common multiplicative factor. This reduces the fraction to a simpler form.
• Finally, make sure no more common factors exist, ensuring the expression is completely simplified to:\[ rac{2}{(x+1)(x-9)} \]

Canceling terms helps in reaching the most reduced form quickly, which is essential for solving algebraic problems efficiently. It's crucial always to ensure that canceled terms are actual factors and not just part of an addition or subtraction, respecting the structure of the expression.