How do you write rational expressions in simplest form?

Question

im having trouble with these three problems:
2 - x + 2x^2 - x^6/ x^2 - 4 

x^2 - 9/ x^3 + X^2 - 9x -9

z^3 - 8/ z^2 + 2z + 4

Answer 1

= (x² - 9)/( x³ + x² - 9x -9)

Consider ( x³ + x² - 9x -9)

Take out common terms x² and 9

x²(x + 1) - 9 (x + 1)

Take out common factors.

( x²- 9 )(x + 1)

There fore

(x² - 9) / ( x³ + x² - 9x -9) = (x² - 9) / [( x²- 9 )(x + 1)]

Simplify

(x² - 9) / ( x³ + x² - 9x -9) = 1 / (x + 1)

Answer 2

= (z³ - 8) / (z² + 2z + 4)

Principal algebraic expression and formula: a³ - b³ = (a - b)(a² + ab + b²)

Consider: z³ - 8 = z³ - 2³ = (z - 2)(z² + 2z + 2²) = (z - 2)(z² + 2z + 4)

(z³ - 8) / (z² + 2z + 4) = [(z - 2)(z² + 2z + 4)] / [z² + 2z +4]

Simplify

= z - 2

There fore

(z³ - 8) / (z² + 2z + 4) = z - 2