3.2 Simplify by factorization: \( \sqrt{x^{2}-8 x+16} \) 3.3 Simplify the following expression: \[ \frac{2^{x}}{2^{x+2}-2^{x+1}} \] 3.4 Simplify the following fractions: 3.4.1 \( \frac{x^{2}-4}{x-2} \) 3.4.2 \( \frac{2}{x^{2}-1}+\frac{x}{x+1}-\frac{3}{x} \)

Factor the expression by following steps: - step0: Factor: \(x^{2}-8x+16\) - step1: Factor the expression: \(\left(x-4\right)^{2}\) Calculate or simplify the expression \( \frac{2^{x}}{2^{x+2}-2^{x+1}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{x}}{2^{x+2}-2^{x+1}}\) - step1: Subtract the terms: \(\frac{2^{x}}{2^{x+1}}\) - step2: Calculate: \(2^{x-\left(x+1\right)}\) - step3: Calculate: \(2^{-1}\) - step4: Express with a positive exponent: \(\frac{1}{2}\) Factor the expression \( x^{2}-4 \). Factor the expression by following steps: - step0: Factor: \(x^{2}-4\) - step1: Rewrite the expression: \(x^{2}-2^{2}\) - step2: Factor the expression: \(\left(x-2\right)\left(x+2\right)\) Calculate or simplify the expression \( \frac{2}{x^{2}-1}+\frac{x}{x+1}-\frac{3}{x} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{x^{2}-1}+\frac{x}{x+1}-\frac{3}{x}\) - step1: Factor the expression: \(\frac{2}{\left(x-1\right)\left(x+1\right)}+\frac{x}{x+1}-\frac{3}{x}\) - step2: Reduce fractions to a common denominator: \(\frac{2x}{\left(x-1\right)\left(x+1\right)x}+\frac{x\left(x-1\right)x}{\left(x+1\right)\left(x-1\right)x}-\frac{3\left(x+1\right)\left(x-1\right)}{x\left(x+1\right)\left(x-1\right)}\) - step3: Rewrite the expression: \(\frac{2x}{\left(x-1\right)\left(x+1\right)x}+\frac{x\left(x-1\right)x}{\left(x-1\right)\left(x+1\right)x}-\frac{3\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)x}\) - step4: Transform the expression: \(\frac{2x+x\left(x-1\right)x-3\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)x}\) - step5: Multiply the terms: \(\frac{2x+x^{3}-x^{2}-3\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)x}\) - step6: Multiply the terms: \(\frac{2x+x^{3}-x^{2}-\left(3x^{2}-3\right)}{\left(x-1\right)\left(x+1\right)x}\) - step7: Calculate: \(\frac{2x+x^{3}-4x^{2}+3}{\left(x-1\right)\left(x+1\right)x}\) - step8: Calculate: \(\frac{2x+x^{3}-4x^{2}+3}{x^{3}-x}\) Calculate or simplify the expression \( \frac{2x+x^{3}-4x^{2}+3}{x^{3}-x} \). Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{2x+x^{3}-4x^{2}+3}{x^{3}-x}\) - step1: Calculate: \(1+\frac{-4x^{2}+3x+3}{x^{3}-x}\) Factor the expression \( x^{3}-x \). Factor the expression by following steps: - step0: Factor: \(x^{3}-x\) - step1: Factor the expression: \(x\left(x^{2}-1\right)\) - step2: Factor the expression: \(x\left(x-1\right)\left(x+1\right)\) Let's simplify each of the expressions step by step. ### 3.2 Simplify by factorization: \( \sqrt{x^{2}-8 x+16} \) 1. The expression inside the square root is \( x^{2}-8x+16 \). 2. We can factor this expression: \[ x^{2}-8x+16 = (x-4)^{2} \] 3. Therefore, the square root simplifies to: \[ \sqrt{x^{2}-8x+16} = \sqrt{(x-4)^{2}} = |x-4| \] ### 3.3 Simplify the following expression: \[ \frac{2^{x}}{2^{x+2}-2^{x+1}} \] 1. Start by simplifying the denominator: \[ 2^{x+2} - 2^{x+1} = 2^{x}(2^{2} - 2^{1}) = 2^{x}(4 - 2) = 2^{x} \cdot 2 = 2^{x+1} \] 2. Now substitute this back into the expression: \[ \frac{2^{x}}{2^{x+1}} = \frac{1}{2} \] ### 3.4 Simplify the following fractions: #### 3.4.1 \( \frac{x^{2}-4}{x-2} \) 1. The expression \( x^{2}-4 \) can be factored as: \[ x^{2}-4 = (x-2)(x+2) \] 2. Therefore, the fraction simplifies to: \[ \frac{(x-2)(x+2)}{x-2} = x+2 \quad \text{(for } x \neq 2\text{)} \] #### 3.4.2 \( \frac{2}{x^{2}-1}+\frac{x}{x+1}-\frac{3}{x} \) 1. First, factor \( x^{2}-1 \): \[ x^{2}-1 = (x-1)(x+1) \] 2. Rewrite the expression: \[ \frac{2}{(x-1)(x+1)} + \frac{x}{x+1} - \frac{3}{x} \] 3. To combine these fractions, find a common denominator, which is \( x(x-1)(x+1) \): \[ \frac{2x}{x(x-1)(x+1)} + \frac{x^2(x-1)}{x(x-1)(x+1)} - \frac{3(x-1)(x+1)}{x(x-1)(x+1)} \] 4. Combine the numerators: \[ \frac{2x + x^2(x-1) - 3(x-1)(x+1)}{x(x-1)(x+1)} \] 5. Simplifying the numerator: \[ x^2(x-1) = x^3 - x^2 \] \[ -3(x-1)(x+1) = -3(x^2 - 1) = -3x^2 + 3 \] Thus, the numerator becomes: \[ 2x + x^3 - x^2 - 3x^2 + 3 = x^3 - 4x^2 + 2x + 3 \] 6. Therefore, the final expression is: \[ \frac{x^3 - 4x^2 + 2x + 3}{x(x-1)(x+1)} \] 7. This can be further simplified to: \[ 1 + \frac{-4x^{2}+3x+3}{x^{3}-x} \] ### Summary of Results: - **3.2:** \( |x-4| \) - **3.3:** \( \frac{1}{2} \) - **3.4.1:** \( x + 2 \) (for \( x \neq 2 \)) - **3.4.2:** \( 1 + \frac{-4x^{2}+3x+3}{x^{3}-x} \)