Problem 27 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{22 a^{4} b^{6} c^{7}(a+2)(a-7)}{4 c(a+2)(a-5)} $$

Step-by-Step Solution

Answer

Question: Simplify the rational expression \(\frac{22 a^{4} b^{6} c^{7}(a+2)(a-7)}{4 c(a+2)(a-5)}\). Answer: \(\frac{11 a^{4} b^{6} c^{6}(a-7)}{2 (a-5)}\)

Step 1: Factor the Numerator and Denominator

We should first take note of the terms that are already factored in the given expression: $$ \frac{22 a^{4} b^{6} c^{7}(a+2)(a-7)}{4 c(a+2)(a-5)}. $$ Since there are no other terms present that can be further factored, we can move on to the next step of simplifying the expression.

Step 2: Cancel out Common Terms

Now that we have factored both the numerator and the denominator, we can cancel out the common terms between them. Observe that both parts of the fraction have the following common terms: \(c\) and \((a+2)\). Since these terms are common between the numerator and the denominator, we can cancel them out: $$ \frac{22 a^{4} b^{6} c^{7}(a+2)(a-7)}{4 c(a+2)(a-5)} \div \frac{c(a+2)}{c(a+2)}. $$ Now after canceling out the common terms, we have: $$ \frac{22 a^{4} b^{6} c^{6}(a-7)}{4 (a-5)}. $$

Step 3: Simplify the Coefficients

Finally, we should simplify the coefficients (the numbers in front of the terms). In this case, we have 22 on the numerator and 4 on the denominator. The greatest common divisor of 22 and 4 is 2; hence, we can simplify the coefficients as follows: $$ \frac{22}{4} = \frac{11\times 2}{2\times 2} = \frac{11}{2}. $$ Now, substitute the simplified coefficients back into the expression: $$ \frac{11 a^{4} b^{6} c^{6}(a-7)}{2 (a-5)}. $$ Therefore, the given rational expression reduced to its lowest terms is: $$ \frac{11 a^{4} b^{6} c^{6}(a-7)}{2 (a-5)}. $$

Key Concepts

Factoring

In the world of rational expressions, factoring is your best friend. It refers to writing a polynomial as a product of its simplest factors, which makes it much easier to work with. When a polynomial expresses something complicated like a sum or a difference, factoring breaks it down into a product—kind of like unscrewing the pieces of a puzzle.
For example, consider the numerator of our rational expression: \(22 a^{4} b^{6} c^{7}(a+2)(a-7)\). Here, it's already broken down into manageable parts, hence it's already factored. Recognizing these parts helps you grasp how they interact with components of the denominator, which is \(4 c(a+2)(a-5)\).
Both the numerator and the denominator contain terms that can't be further factored (like primes, they are in their simplest form). If any part was missing or incorrectly represented, factoring would guide us to correct it. Once fully factored, you can spot and cancel common terms in the next step.

Canceling Common Terms

After you factor both the numerator and the denominator, the next exciting step is canceling out the common terms. It's like spring cleaning your equation — making things neat and tidy.
In our example, you notice both the numerator and the denominator have a \(c\) and \((a+2)\).

1. The \(c\) represents a simple factor common in both, contributing to the 'clean-up'.
2. The term \((a+2)\) is a binomial common to both parts. Canceling out both transforms our equation significantly.

The result of this canceling operation is a much simpler expression \(\frac{22 a^{4} b^{6} c^{6}(a-7)}{4 (a-5)}\). This step exemplifies the power of canceling—it helps in moving us closer to a simplified version without changing the value. Always ensure the terms are indeed common between the two parts before canceling to avoid changing the equation's value by mistake.

Simplifying Coefficients

Simplifying coefficients is the grand finale of reducing a rational expression. It's the process of taking numerical coefficients and making them as simple as possible. In our current scenario, on top of \(a^{4} b^{6} c^{6}(a-7)\), we have the numbers 22 and 4.
The secret here is to use the greatest common divisor (GCD). The GCD of 22 and 4 happens to be 2. This means both numbers can be divided by 2, leading to much simpler fractions. So we deduce:

• Divide 22 by 2 giving 11.
• Divide 4 by 2 giving 2.

After this crucial step, we update our expression to its simplest form: \(\frac{11 a^{4} b^{6} c^{6}(a-7)}{2 (a-5)}\). This transformation is important not only for making the expression look less intimidating but also for practical applications where simplified forms are easier to work with. Coefficients give us very simplified results that maintain the original equation's integrity.