Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
adding and subtracting rational expressions
explaining how to add rational expressions
put the steps for adding two rational expressions in order.
if possible, simplify by factoring the numerator and dividing out common factors from both the numerator and denominator.
add the numerators, keeping the denominators the same.
factor the denominators of each expression to find the lcd.
rename each expression, if needed, by multiplying by a form of one to get the lcd.

Explanation:

To determine the correct order of steps for adding two rational expressions, we follow the standard procedure for adding fractions (rational expressions are like fractions with polynomials):

Step 1: Factor denominators

First, we factor the denominators of each rational expression. This helps identify the least common denominator (LCD) by finding the product of the highest power of all prime factors (or polynomial factors) present.
Example: For \(\frac{1}{x^2 - 4} + \frac{3}{x + 2}\), factor \(x^2 - 4 = (x - 2)(x + 2)\) and \(x + 2\) remains as is.

Step 2: Rename to get LCD

Next, we rename (rewrite) each rational expression by multiplying numerator and denominator by the missing factors needed to make the denominator equal to the LCD. The LCD here would be \((x - 2)(x + 2)\). For \(\frac{3}{x + 2}\), we multiply numerator and denominator by \((x - 2)\) to get \(\frac{3(x - 2)}{(x - 2)(x + 2)}\).

Step 3: Add numerators (same denominator)

Once the denominators are the same, we add the numerators while keeping the common denominator. Using the example:
\(\frac{1}{(x - 2)(x + 2)} + \frac{3(x - 2)}{(x - 2)(x + 2)} = \frac{1 + 3(x - 2)}{(x - 2)(x + 2)}\).

Step 4: Simplify (if possible)

Finally, we simplify the resulting expression by factoring the numerator (if possible) and dividing out common factors with the denominator. Expanding and simplifying the numerator:
\(1 + 3(x - 2) = 1 + 3x - 6 = 3x - 5\). The denominator is \((x - 2)(x + 2)\), and there are no common factors, so the simplified form is \(\frac{3x - 5}{(x - 2)(x + 2)}\).

Final Order of Steps:

1. Factor the denominators of each expression to find the LCD.
2. Rename each expression, if needed, by multiplying by a form of one to get the LCD.
3. Add the numerators, keeping the denominators the same.
4. If possible, simplify by factoring the numerator and dividing out common factors from both the numerator and denominator.

Answer:

To determine the correct order of steps for adding two rational expressions, we follow the standard procedure for adding fractions (rational expressions are like fractions with polynomials):

Step 1: Factor denominators

First, we factor the denominators of each rational expression. This helps identify the least common denominator (LCD) by finding the product of the highest power of all prime factors (or polynomial factors) present.
Example: For \(\frac{1}{x^2 - 4} + \frac{3}{x + 2}\), factor \(x^2 - 4 = (x - 2)(x + 2)\) and \(x + 2\) remains as is.

Step 2: Rename to get LCD

Next, we rename (rewrite) each rational expression by multiplying numerator and denominator by the missing factors needed to make the denominator equal to the LCD. The LCD here would be \((x - 2)(x + 2)\). For \(\frac{3}{x + 2}\), we multiply numerator and denominator by \((x - 2)\) to get \(\frac{3(x - 2)}{(x - 2)(x + 2)}\).

Step 3: Add numerators (same denominator)

Once the denominators are the same, we add the numerators while keeping the common denominator. Using the example:
\(\frac{1}{(x - 2)(x + 2)} + \frac{3(x - 2)}{(x - 2)(x + 2)} = \frac{1 + 3(x - 2)}{(x - 2)(x + 2)}\).

Step 4: Simplify (if possible)

Finally, we simplify the resulting expression by factoring the numerator (if possible) and dividing out common factors with the denominator. Expanding and simplifying the numerator:
\(1 + 3(x - 2) = 1 + 3x - 6 = 3x - 5\). The denominator is \((x - 2)(x + 2)\), and there are no common factors, so the simplified form is \(\frac{3x - 5}{(x - 2)(x + 2)}\).

Final Order of Steps:

1. Factor the denominators of each expression to find the LCD.
2. Rename each expression, if needed, by multiplying by a form of one to get the LCD.
3. Add the numerators, keeping the denominators the same.
4. If possible, simplify by factoring the numerator and dividing out common factors from both the numerator and denominator.