Question

simplify the following rational expressions:
\\(\frac{x + 3}{x^2 - 2x + 1} + \frac{x}{x^2 - 3x + 2}\\)
\\(\frac{x}{x^2 - 4x + 4} - \frac{2}{x^2 - 4}\\)

Explanation:

To solve the given rational expressions, we'll simplify each part by factoring the denominators and then combining the fractions. Let's start with the first expression: $\boldsymbol{\frac{x + 3}{x^2 - 2x + 1} + \frac{x}{x^2 - 3x + 2}}$

Step 1: Factor the Denominators

• For $x^2 - 2x + 1$: This is a perfect square trinomial.

$x^2 - 2x + 1 = (x - 1)^2$

• For $x^2 - 3x + 2$: Factor by finding two numbers that multiply to $2$ and add to $-3$.

$x^2 - 3x + 2 = (x - 1)(x - 2)$

Step 2: Find the Least Common Denominator (LCD)

The denominators are $(x - 1)^2$ and $(x - 1)(x - 2)$. The LCD is the product of the highest powers of all unique factors:
$\text{LCD} = (x - 1)^2(x - 2)$

Step 3: Rewrite Fractions with the LCD

• For $\frac{x + 3}{(x - 1)^2}$: Multiply numerator and denominator by $(x - 2)$ to get the LCD.

$\frac{x + 3}{(x - 1)^2} \cdot \frac{x - 2}{x - 2} = \frac{(x + 3)(x - 2)}{(x - 1)^2(x - 2)}$

• For $\frac{x}{(x - 1)(x - 2)}$: Multiply numerator and denominator by $(x - 1)$ to get the LCD.

$\frac{x}{(x - 1)(x - 2)} \cdot \frac{x - 1}{x - 1} = \frac{x(x - 1)}{(x - 1)^2(x - 2)}$

Step 4: Add the Fractions

Combine the numerators over the common denominator:

$$\begin{align*} \frac{(x + 3)(x - 2) + x(x - 1)}{(x - 1)^2(x - 2)} &= \frac{(x^2 + x - 6) + (x^2 - x)}{(x - 1)^2(x - 2)} \\ &= \frac{x^2 + x - 6 + x^2 - x}{(x - 1)^2(x - 2)} \\ &= \frac{2x^2 - 6}{(x - 1)^2(x - 2)} \\ &= \frac{2(x^2 - 3)}{(x - 1)^2(x - 2)} \quad \text{(Factor out 2)} \end{align*}$$

Now, let's simplify the second expression: $\boldsymbol{\frac{x}{x^2 - 4x + 4} - \frac{2}{x^2 - 4}}$

Step 1: Factor the Denominators

• For $x^2 - 4x + 4$: This is a perfect square trinomial.

$x^2 - 4x + 4 = (x - 2)^2$

• For $x^2 - 4$: This is a difference of squares.

$x^2 - 4 = (x - 2)(x + 2)$

Step 2: Find the Least Common Denominator (LCD)

The denominators are $(x - 2)^2$ and $(x - 2)(x + 2)$. The LCD is:
$\text{LCD} = (x - 2)^2(x + 2)$

Step 3: Rewrite Fractions with the LCD

• For $\frac{x}{(x - 2)^2}$: Multiply numerator and denominator by $(x + 2)$ to get the LCD.

$\frac{x}{(x - 2)^2} \cdot \frac{x + 2}{x + 2} = \frac{x(x + 2)}{(x - 2)^2(x + 2)}$

• For $\frac{2}{(x - 2)(x + 2)}$: Multiply numerator and denominator by $(x - 2)$ to get the LCD.

$\frac{2}{(x - 2)(x + 2)} \cdot \frac{x - 2}{x - 2} = \frac{2(x - 2)}{(x - 2)^2(x + 2)}$

Step 4: Subtract the Fractions

Combine the numerators over the common denominator:

$$\begin{align*} \frac{x(x + 2) - 2(x - 2)}{(x - 2)^2(x + 2)} &= \frac{(x^2 + 2x) - (2x - 4)}{(x - 2)^2(x + 2)} \\ &= \frac{x^2 + 2x - 2x + 4}{(x - 2)^2(x + 2)} \\ &= \frac{x^2 + 4}{(x - 2)^2(x + 2)} \end{align*}$$

Final Simplified Expressions

• First expression: $\boldsymbol{\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}}$
• Second expression: $\boldsymbol{\frac{x^2 + 4}{(x - 2)^2(x + 2)}}$

(Note: If you need to combine these two expressions further, additional factoring or simplification may be required, but the problem does not specify combining them. The above steps simplify each rational expression separately.)

Answer:

To solve the given rational expressions, we'll simplify each part by factoring the denominators and then combining the fractions. Let's start with the first expression: $\boldsymbol{\frac{x + 3}{x^2 - 2x + 1} + \frac{x}{x^2 - 3x + 2}}$

Step 1: Factor the Denominators

• For $x^2 - 2x + 1$: This is a perfect square trinomial.

$x^2 - 2x + 1 = (x - 1)^2$

• For $x^2 - 3x + 2$: Factor by finding two numbers that multiply to $2$ and add to $-3$.

$x^2 - 3x + 2 = (x - 1)(x - 2)$

Step 2: Find the Least Common Denominator (LCD)

The denominators are $(x - 1)^2$ and $(x - 1)(x - 2)$. The LCD is the product of the highest powers of all unique factors:
$\text{LCD} = (x - 1)^2(x - 2)$

Step 3: Rewrite Fractions with the LCD

• For $\frac{x + 3}{(x - 1)^2}$: Multiply numerator and denominator by $(x - 2)$ to get the LCD.

$\frac{x + 3}{(x - 1)^2} \cdot \frac{x - 2}{x - 2} = \frac{(x + 3)(x - 2)}{(x - 1)^2(x - 2)}$

• For $\frac{x}{(x - 1)(x - 2)}$: Multiply numerator and denominator by $(x - 1)$ to get the LCD.

$\frac{x}{(x - 1)(x - 2)} \cdot \frac{x - 1}{x - 1} = \frac{x(x - 1)}{(x - 1)^2(x - 2)}$

Step 4: Add the Fractions

Combine the numerators over the common denominator:

$$\begin{align*} \frac{(x + 3)(x - 2) + x(x - 1)}{(x - 1)^2(x - 2)} &= \frac{(x^2 + x - 6) + (x^2 - x)}{(x - 1)^2(x - 2)} \\ &= \frac{x^2 + x - 6 + x^2 - x}{(x - 1)^2(x - 2)} \\ &= \frac{2x^2 - 6}{(x - 1)^2(x - 2)} \\ &= \frac{2(x^2 - 3)}{(x - 1)^2(x - 2)} \quad \text{(Factor out 2)} \end{align*}$$

Now, let's simplify the second expression: $\boldsymbol{\frac{x}{x^2 - 4x + 4} - \frac{2}{x^2 - 4}}$

Step 1: Factor the Denominators

• For $x^2 - 4x + 4$: This is a perfect square trinomial.

$x^2 - 4x + 4 = (x - 2)^2$

• For $x^2 - 4$: This is a difference of squares.

$x^2 - 4 = (x - 2)(x + 2)$

Step 2: Find the Least Common Denominator (LCD)

The denominators are $(x - 2)^2$ and $(x - 2)(x + 2)$. The LCD is:
$\text{LCD} = (x - 2)^2(x + 2)$

Step 3: Rewrite Fractions with the LCD

• For $\frac{x}{(x - 2)^2}$: Multiply numerator and denominator by $(x + 2)$ to get the LCD.

$\frac{x}{(x - 2)^2} \cdot \frac{x + 2}{x + 2} = \frac{x(x + 2)}{(x - 2)^2(x + 2)}$

• For $\frac{2}{(x - 2)(x + 2)}$: Multiply numerator and denominator by $(x - 2)$ to get the LCD.

$\frac{2}{(x - 2)(x + 2)} \cdot \frac{x - 2}{x - 2} = \frac{2(x - 2)}{(x - 2)^2(x + 2)}$

Step 4: Subtract the Fractions

Combine the numerators over the common denominator:

$$\begin{align*} \frac{x(x + 2) - 2(x - 2)}{(x - 2)^2(x + 2)} &= \frac{(x^2 + 2x) - (2x - 4)}{(x - 2)^2(x + 2)} \\ &= \frac{x^2 + 2x - 2x + 4}{(x - 2)^2(x + 2)} \\ &= \frac{x^2 + 4}{(x - 2)^2(x + 2)} \end{align*}$$

Final Simplified Expressions

• First expression: $\boldsymbol{\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}}$
• Second expression: $\boldsymbol{\frac{x^2 + 4}{(x - 2)^2(x + 2)}}$

(Note: If you need to combine these two expressions further, additional factoring or simplification may be required, but the problem does not specify combining them. The above steps simplify each rational expression separately.)