Problem 55 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{6 b-6}{-3}\)

Step-by-Step Solution

Answer

Answer: The simplest form of the rational expression is \(-2b+2\).

Step 1: Factor both the numerator and the denominator

In this step, factor out any common factors in both the numerator and the denominator. For the numerator, we can factor out the common factor of 6. For the denominator, we can't do any further factoring as -3 is a prime number. So we get: \(\frac{6(b-1)}{-3}\)

Step 2: Simplify the rational expression by cancelling out the common factors

Now we can simplify the expression by dividing out the common factors from both the numerator and the denominator. The common factor is 3, so we can divide both the numerator and the denominator by 3. \(\frac{6}{-3} (b-1)\) The division yields the result: \(-2(b-1)\)

Step 3: Write the final expression

Now the expression is reduced to its lowest terms. We can rewrite it as: \(-2b+2\) The simplified rational expression is \(-2b+2\).

Key Concepts

Factoring

Factoring plays a crucial role in simplifying rational expressions. It involves rewriting a number or expression as a product of its factors. In our original exercise, we started by factoring the numerator, which was \(6b - 6\). Both terms in the expression share a common factor of 6. Factoring out the 6, the expression becomes \(6(b - 1)\).

• Factoring Tips: Look for the greatest common factor (GCF) in the terms. Factor it out, which simplifies the expression.
• Resist the urge to skip the factoring step even if it seems simple. It is foundational to further simplification.

In the exercise, the denominator -3 is a prime number, meaning it cannot be factored further. So, we keep the denominator as it is while factoring.

Simplifying Expressions

Once you've factored both the numerator and the denominator, the next step is simplifying the expression by canceling out common factors. This is crucial because it transforms the expression into a simpler form.

Cancel Common Factors

To simplify \(\frac{6(b-1)}{-3}\), observe that both 6 and -3 have a common factor of 3. By canceling out this common factor, we divide both terms by 3:

• The term 6 in the numerator becomes 2.
• The denominator -3 becomes -1.

Thus, the expression simplifies to \(-2(b-1)\). Simplifying makes the expression easier to understand and work with.

Lowest Terms

Reducing an expression to its lowest terms means it cannot be simplified further. This is your goal for any rational expression. You already canceled out the common factors in the previous steps, leaving the expression \(-2(b-1)\). Expanding this back in standard algebraic form, you obtain \(-2b + 2\).

Check Your Work

• Verify that all common factors between the numerator and the denominator have been canceled.
• Ensure that any negative signs are also handled correctly for clear and correct final expression.

Finally, always double-check your expression to make sure it’s expressed in the simplest form possible, confirming it’s truly at its lowest terms.