Simplify the following expressions: (x^6-1)*1/(x^3+1) *(x+1)/(x^2+x+1) and x/(x^2-6x+9) - (x+5)/(x^2+2x-15)

Let's simplify the expressions step by step:

Expression 1:

\[ (x^6-1) \cdot \frac{1}{x^3+1} \cdot \frac{x+1}{x^2+x+1} \] We know that \( a^2 - b^2 = (a - b)(a + b) \) and \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). 1. Factor \( x^6 - 1 \) as a difference of squares: \[ x^6 - 1 = (x^3)^2 - 1^2 = (x^3 - 1)(x^3 + 1) \] 2. Factor \( x^3 - 1 \) as a difference of cubes: \[ x^3 - 1 = (x - 1)(x^2 + x + 1) \] So, \( x^6 - 1 = (x - 1)(x^2 + x + 1)(x^3 + 1) \) 3. The expression becomes: \[ (x - 1)(x^2 + x + 1)(x^3 + 1) \cdot \frac{1}{x^3+1} \cdot \frac{x+1}{x^2+x+1} \] 4. Cancel out common terms: \[ (x - 1) \cdot (x+1) \] 5. Multiply the remaining terms: \[ (x - 1)(x + 1) = x^2 - 1 \] Thus, the simplified form of the first expression is: \[ x^2 - 1 \]

Expression 2:

\[ \frac{x}{x^2-6x+9} - \frac{x+5}{x^2+2x-15} \] 1. Factor the denominators: \[ x^2 - 6x + 9 = (x - 3)^2 \] \[ x^2 + 2x - 15 = (x + 5)(x - 3) \] 2. The expression becomes: \[ \frac{x}{(x - 3)^2} - \frac{x+5}{(x + 5)(x - 3)} \] 3. Cancel out common terms: \[ \frac{x}{(x - 3)^2} - \frac{1}{x - 3} \] 4. Find a common denominator, which is \( (x - 3)^2 \): \[ \frac{x}{(x - 3)^2} - \frac{x - 3}{(x - 3)^2} \] 5. Combine the fractions: \[ \frac{x - (x - 3)}{(x - 3)^2} \] 6. Simplify the numerator: \[ \frac{x - x + 3}{(x - 3)^2} = \frac{3}{(x - 3)^2} \] Thus, the simplified form of the second expression is: \[ \frac{3}{(x - 3)^2} \]