Adding and Subtracting Rational Expressions

This lesson focuses on adding and subtracting rational expressions with unlike denominators. The primary goal is to obtain common denominators before combining the fractions. Here are the detailed steps and examples provided:

General Approach:

1. Before combining fractions, it is essential to get common denominators.

2. The least common multiple (LCM) of the denominators can be used as the common denominator. Alternatively, you can simply multiply the denominators together, though this might require more simplification later.

Examples:

1. Example 1: $\frac{5}{x} + \frac{3}{x^2}$

   • Problem: Add $\frac{5}{x}$ and $\frac{3}{x^2}$.

   • Step 1: Identify the common denominator. The denominators are $x$ and $x^2$. The common denominator needed is $x^2$.

   • Step 2: Adjust the first fraction. To make the denominator of the first fraction ($\frac{5}{x}$) equal to $x^2$, multiply it by $\frac{x}{x}$. This results in $\frac{5 \cdot x}{x \cdot x} = \frac{5x}{x^2}$.

   • Step 3: Rewrite the expression. The expression now becomes $\frac{5x}{x^2} + \frac{3}{x^2}$.

   • Step 4: Combine the fractions. Since the denominators are the same, add the numerators: $\frac{5x + 3}{x^2}$.

   • Answer: The simplified sum is $\frac{5x + 3}{x^2}$.

2. Example 2: $\frac{x - 3}{4} - \frac{x + 2}{3}$

   • Problem: Subtract $\frac{x + 2}{3}$ from $\frac{x - 3}{4}$.

   • Step 1: Identify the common denominator. The denominators are $4$ and $3$. The LCM of $4$ and $3$ is 12.

   • Step 2: Adjust the first fraction. Multiply the first fraction ($\frac{x - 3}{4}$) by $\frac{3}{3}$. This gives $\frac{3(x - 3)}{4 \cdot 3} = \frac{3(x - 3)}{12}$.

   • Step 3: Adjust the second fraction. Multiply the second fraction ($\frac{x + 2}{3}$) by $\frac{4}{4}$. This gives $\frac{4(x + 2)}{3 \cdot 4} = \frac{4(x + 2)}{12}$.

   • Step 4: Distribute in the numerators.

     • First numerator: $3(x - 3) = 3x - 9$.

     • Second numerator: $4(x + 2) = 4x + 8$.

   • Step 5: Rewrite the expression. The expression now is $\frac{3x - 9}{12} - \frac{4x + 8}{12}$.

   • Step 6: Combine the fractions. Write as a single fraction, being careful to distribute the negative sign to both terms in the second numerator:
     $\frac{(3x - 9) - (4x + 8)}{12}$ becomes $\frac{3x - 9 - 4x - 8}{12}$.

   • Step 7: Combine like terms in the numerator.

     • Combine $x$ terms: $3x - 4x = -x$.

     • Combine constant terms: $-9 - 8 = -17$.

   • Answer: The simplified difference is $\frac{-x - 17}{12}$.

   • Alternative Formatting: You can factor out a negative sign from the numerator: $\frac{-(x + 17)}{12}$. The negative sign can also be placed in front of the fraction: $-\frac{x + 17}{12}$.

3. Example 3: $\frac{4}{x - 2} + \frac{5}{x + 2}$

   • Problem: Add $\frac{4}{x - 2}$ and $\frac{5}{x + 2}$.

   • Step 1: Identify the common denominator. The denominators are $(x - 2)$ and $(x + 2)$. Multiply these two unique denominators to get the common denominator: $(x - 2)(x + 2)$.

   • Step 2: Adjust the first fraction. Multiply the first fraction ($\frac{4}{x - 2}$) by $\frac{x + 2}{x + 2}$. This gives $\frac{4(x + 2)}{(x - 2)(x + 2)}$.

   • Step 3: Adjust the second fraction. Multiply the second fraction ($\frac{5}{x + 2}$) by $\frac{x - 2}{x - 2}$. This gives $\frac{5(x - 2)}{(x - 2)(x + 2)}$.

   • Step 4: Distribute in the numerators.

     • First numerator: $4(x + 2) = 4x + 8$.

     • Second numerator: $5(x - 2) = 5x - 10$.

   • Step 5: Simplify the common denominator (optional but good practice). The common denominator $(x - 2)(x + 2)$ is a difference of squares, which simplifies to $x^2 - 4$.

   • Step 6: Combine the numerators. Since the denominators are now the same, add the numerators:
     $(4x + 8) + (5x - 10)$.

   • Step 7: Combine like terms.

     • Combine $x$ terms: $4x + 5x = 9x$.

     • Combine constant terms: $8 + (-10) = -2$.

   • Answer: The simplified sum is $\frac{9x - 2}{x^2 - 4}$. This can also be written as $\frac{9x - 2}{(x - 2)(x + 2)}$.

4. Example 4: $\frac{x}{x^2 + 9x + 20} - \frac{4}{x^2 + 7x + 12}$

   • Problem: Subtract $\frac{4}{x^2 + 7x + 12}$ from $\frac{x}{x^2 + 9x + 20}$.

   • Step 1: Factor completely the denominators.

     • For $x^2 + 9x + 20$: Find two numbers that multiply to $20$ and add to $9$. These are $4$ and $5$. So, $x^2 + 9x + 20 = (x + 4)(x + 5)$.

     • For $x^2 + 7x + 12$: Find two numbers that multiply to $12$ and add to $7$. These are $3$ and $4$. So, $x^2 + 7x + 12 = (x + 3)(x + 4)$.

   • Step 2: Rewrite the expression with factored denominators.
     $\frac{x}{(x + 4)(x + 5)} - \frac{4}{(x + 3)(x + 4)}$.

   • Step 3: Identify the common denominator. Both fractions already share the factor $(x + 4)$. To form the LCM, multiply all unique factors together:

     • The common denominator will be $(x + 3)(x + 4)(x + 5)$.

   • Step 4: Adjust each fraction to have the common denominator.

     • First fraction ($\frac{x}{(x + 4)(x + 5)}$): Multiply by $\frac{(x + 3)}{(x + 3)}$. This gives $\frac{x(x + 3)}{(x + 3)(x + 4)(x + 5)}$.

     • Second fraction ($\frac{4}{(x + 3)(x + 4)}$): Multiply by $\frac{(x + 5)}{(x + 5)}$. This gives $\frac{4(x + 5)}{(x + 3)(x + 4)(x + 5)}$.

   • Step 5: Distribute in the numerators.

     • First numerator: $x(x + 3) = x^2 + 3x$.

     • Second numerator: Remember the negative sign from the subtraction. $-4(x + 5) = -4x - 20$.

   • Step 6: Combine the numerators. Write as a single fraction over the common denominator:
     $\frac{(x^2 + 3x) + (-4x - 20)}{(x + 3)(x + 4)(x + 5)} = \frac{x^2 + 3x - 4x - 20}{(x + 3)(x + 4)(x + 5)}$.

   • Step 7: Combine like terms in the numerator.

     • Combine $x$ terms: $3x - 4x = -x$.

     • The numerator becomes $x^2 - x - 20$.

   • Step 8: Further Factorization of the Numerator (if possible).

     • Can $x^2 - x - 20$ be factored? Find two numbers that multiply to $-20$ and add to $-1$. These numbers are $-5$ and $4$.

     • So, the numerator factors to $(x - 5)(x + 4)$.

   • Step 9: Rewrite the expression with the factored numerator.
     $\frac{(x - 5)(x + 4)}{(x + 3)(x + 4)(x + 5)}$.

   • Step 10: Cancellation. Notice that $(x + 4)$ is a common factor in both the numerator and the denominator. Cancel out $(x + 4)$.

   • Final Answer: The simplified expression is $\frac{x - 5}{(x + 3)(x + 5)}$.

5. Example 5: $\frac{x^2}{x - 4} + \frac{7}{4 - x}$

   • Problem: Add $\frac{x^2}{x - 4}$ and $\frac{7}{4 - x}$.

   • Step 1: Recognize special case: Denominators are opposites. The denominators $(x - 4)$ and $(4 - x)$ are opposites.

   • Step 2: Obtain a common denominator by factoring out -1. Factor out $-1$ from one of the denominators. For example, from $(4 - x)$:
     $4 - x = -1(x - 4)$.

   • Step 3: Rewrite the second fraction. Substitute the factored denominator into the second fraction:
     $\frac{7}{-1(x - 4)}$.

   • Step 4: Move the negative sign. Move the negative sign from the denominator to the numerator (or in front of the fraction):
     $\frac{-7}{x - 4}$.

   • Step 5: Rewrite the entire expression. The expression now becomes $\frac{x^2}{x - 4} - \frac{7}{x - 4}$.

   • Step 6: Combine the fractions. Since the denominators are now the same, combine the numerators:
     $\frac{x^2 - 7}{x - 4}$.

   • Answer: The simplified sum is $\frac{x^2 - 7}{x - 4}$.

6. Example 6: $\frac{5}{x + 2} + \frac{2}{x + 1} - \frac{3}{x - 1}$

   • Problem: Add and subtract three rational expressions.

   • Step 1: Identify the common denominator. Multiply all three unique denominators to get the common denominator:
     $(x + 2)(x + 1)(x - 1)$.

   • Step 2: Adjust each fraction to have the common denominator. (Multiply each fraction by the other two denominators' factors).

     • First fraction ($\frac{5}{x + 2}$): Multiply by $\frac{(x + 1)(x - 1)}{(x + 1)(x - 1)}$.

     • Second fraction ($\frac{2}{x + 1}$): Multiply by $\frac{(x + 2)(x - 1)}{(x + 2)(x - 1)}$.

     • Third fraction ($\frac{3}{x - 1}$): Multiply by $\frac{(x + 2)(x + 1)}{(x + 2)(x + 1)}$.

   • Step 3: Expand each numerator.

     • For the first numerator: $5 imes ((x + 1)(x - 1))$. First, FOIL $(x + 1)(x - 1)$ to get $x^2 - 1$. Then distribute $5$: $5(x^2 - 1) = 5x^2 - 5$.

     • For the second numerator: $2 imes ((x + 2)(x - 1))$. First, FOIL $(x + 2)(x - 1)$ to get $x^2 + x - 2$. Then distribute $2$: $2(x^2 + x - 2) = +2x^2 + 2x - 4$.

     • For the third numerator: Remember the negative sign. $-3 imes ((x + 2)(x + 1))$. First, FOIL $(x + 2)(x + 1)$ to get $x^2 + 3x + 2$. Then distribute $-3$: $-3(x^2 + 3x + 2) = -3x^2 - 9x - 6$.

   • Step 4: Combine all expanded numerator terms over the common denominator.
     $\frac{(5x^2 - 5) + (2x^2 + 2x - 4) + (-3x^2 - 9x - 6)}{(x + 2)(x + 1)(x - 1)}$.

   • Step 5: Combine like terms in the numerator.

     • Combine $x^2$ terms: $5x^2 + 2x^2 - 3x^2 = 4x^2$.

     • Combine $x$ terms: $2x - 9x = -7x$.

     • Combine constant terms: $-5 - 4 - 6 = -15$.

   • Step 6: State the combined numerator. The combined numerator is $4x^2 - 7x - 15$.

   • Step 7: Attempt to Factor the Numerator.

     • We need to factor $4x^2 - 7x - 15$. Use the AC method: multiply $a imes c$: $4 imes (-15) = -60$.

     • Find two numbers that multiply to $-60$ and add to $-7$. These numbers are $5$ and $-12$ ($5 imes -12 = -60$ and $5 + (-12) = -7$).

     • Replace the middle term $-7x$ with $-12x + 5x$: $4x^2 - 12x + 5x - 15$.

     • Factor by grouping:

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