Problem 90 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{x-2}{x^{2}+7 x+6}, \frac{2 x}{x^{2}+4 x-12} $$

Step-by-Step Solution

Answer

Question: Rewrite the following rational expressions with the least common denominator: $$ \frac{x - 2}{x^2 + 7x + 6}, \frac{2x}{x^2 + 4x - 12}. $$ Answer: $$ \frac{(x - 2)^2}{(x + 1)(x - 2)(x + 6)}, \frac{2x(x + 1)}{(x + 1)(x - 2)(x + 6)}. $$

Step 1: Factor the denominators

First, we need to factor the denominators of the two rational expressions: $$ x^2 + 7x + 6 = (x + 1)(x + 6) \\ x^2 + 4x - 12 = (x + 6)(x - 2). $$

Step 2: Find the least common multiple (LCM) of the denominators

Now that we have factored the denominators, we can identify the least common multiple (LCM) as: $$ \text{LCM}((x + 1)(x + 6), (x + 6)(x - 2)) = (x + 1)(x - 2)(x + 6). $$

Step 3: Rewrite each rational expression with the LCD

With the least common denominator found, we can now rewrite each rational expression, making sure the numerator is adjusted accordingly to maintain the same value. For the first expression, we need to multiply the numerator and denominator by \((x - 2)\): $$ \frac{x - 2}{(x + 1)(x + 6)} \cdot \frac{x - 2}{x - 2} = \frac{(x - 2)^2}{(x + 1)(x - 2)(x + 6)}. $$ For the second expression, we need to multiply the numerator and denominator by \((x + 1)\): $$ \frac{2x}{(x + 6)(x - 2)} \cdot \frac{x + 1}{x + 1} = \frac{2x(x + 1)}{(x + 1)(x - 2)(x + 6)}. $$

Step 4: Write the final expressions

Now that both rational expressions have the same denominator, we can write the final rational expressions as: $$ \frac{(x - 2)^2}{(x + 1)(x - 2)(x + 6)}, \frac{2x(x + 1)}{(x + 1)(x - 2)(x + 6)}. $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill when working with algebraic expressions, especially when simplifying rational expressions. It essentially involves breaking down a polynomial into a product of simpler polynomials, which are called factors. These factors are easier to work with, especially when adding, subtracting, or finding the least common multiple (LCM) for algebraic fractions.

To factor a polynomial, look for a common factor in each term, and use it to simplify the expression. There are several methods employed in factoring polynomials, including factoring by grouping, using special product formulas (like the difference of squares or perfect square trinomials), or the quadratic formula when dealing with second-degree polynomials. In the exercise provided, the denominator of each fraction is factored into binomials, which simplifies the process of finding a least common denominator (LCD) and reducing the expressions to their simplest form.

Understanding how to effectively factor polynomials is crucial because it's a foundational step that makes the rest of the rational expression simplification process possible.

Least Common Multiple

The least common multiple (LCM) of two or more algebraic expressions is the simplest expression that is a multiple of all the expressions. In the context of simplifying rational expressions, the LCM is used to find the least common denominator (LCD) when adding, subtracting, or comparing fractions with different denominators.

To find the LCM of polynomials, first factor them into their irreducible factors. Then take the highest powers of all the factors that appear in the factored form. For rational expressions, this would translate into having a common bottom, the denominator, so that the expressions can be combined if needed. As shown in the provided solution, after factoring the denominators of two fractions, identifying the LCM is just a matter of combining the unique factors using the highest powers from each expression.

Finding the LCM is crucial not only in algebra but also in many practical applications, such as working with units of measurement or scheduling where different cycles need to be harmonized.

Equivalent Rational Expressions

Equivalent rational expressions are different expressions that represent the same value or function. To create equivalent rational expressions, especially when working with different denominators, one must find a common denominator, which would typically be the LCM of all denominators involved. In the exercise, the two given rational expressions are converted to equivalent forms with the same denominator for the purpose of comparison or possible addition and subtraction.

Manipulating a rational expression to create an equivalent one involves multiplying both the numerator and the denominator by the same expression. This is analogous to multiplying the fraction by one since any number divided by itself is one. This step does not change the value of the rational expression, but it does change its form. In simplifying complex rational expressions, it is important to remember that although the forms may change, the values represented by the expressions remain the same, provided the original expression isn't undefined for some values.

Understanding how to create equivalent rational expressions is pivotal in solving equations, simplifying complex expressions, and functioning within topics such as calculus where algebraic manipulation is a common practice.