Q1. Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (i) (3p² - 3pq - 18q²) / (p² + 3pq - 10q²) (ii) (w³ - v³ + x³ + 3wvx) / (w² + v² + x² - 2wv - 2vx + 2wx) (iii) ((x² + x - 6)(x² - 7x + 12)) / ((x² - 6x + 8)(x² - 9))

Answer:

(i) Numerator: 3p² - 3pq - 18q² = 3(p² - pq - 6q²) = 3(p - 3q)(p + 2q)

Denominator: p² + 3pq - 10q² = (p + 5q)(p - 2q)

So, the expression becomes: 3(p - 3q)(p + 2q) / ((p + 5q)(p - 2q)). There are no common factors, so this is the simplified form.

(ii) Numerator: w³ - v³ + x³ + 3wvx = w³ + (-v)³ + x³ - 3w(-v)x = w³ + (-v)³ + x³ - 3w(-v)x.

Using the identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), with a = w, b = -v, c = x:

w³ + (-v)³ + x³ - 3w(-v)x = (w - v + x)(w² + v² + x² - w(-v) - (-v)x - xw) = (w - v + x)(w² + v² + x² + wv + vx - wx).

Denominator: w² + v² + x² - 2wv - 2vx + 2wx = w² + (-v)² + x² + 2w(-v) + 2(-v)x + 2xw = (w - v + x)².

Thus, the expression simplifies to: (w - v + x)(w² + v² + x² + wv + vx - wx) / (w - v + x)² = (w² + v² + x² + wv + vx - wx) / (w - v + x).

(iii) Factor each quadratic:

x² + x - 6 = (x + 3)(x - 2)

x² - 7x + 12 = (x - 3)(x - 4)

x² - 6x + 8 = (x - 2)(x - 4)

x² - 9 = (x - 3)(x + 3)

Numerator: (x + 3)(x - 2)(x - 3)(x - 4)

Denominator: (x - 2)(x - 4)(x - 3)(x + 3)

Cancelling common factors, we get 1, provided none of the factors is zero.