Question

6.1 multiplying & dividing rational expressions
directions: on #1-3, write each rational expression in lowest terms.

1. \\(\frac{9x^2 + 27x^3}{7x + 21x^2}\\) 2. \\(\frac{x^2 - 3x - 10}{x^2 - 4x - 12}\\) 3. \\(\frac{x^2 + 9x + 18}{36 - x^2}\\)

directions: on #4-7, multiply.

1. \\(\frac{2x - 2}{x} \cdot \frac{3x^2}{9x - 9}\\) 5. \\(\frac{x^2 + 11x + 24}{x^2 + 12x + 27} \cdot \frac{x^2 + 9x}{x^2 + 4x - 32}\\)
2. \\(\frac{x^2 - 4x + 3}{x^2 - 12x + 20} \cdot \frac{x^2 - 19x + 90}{x^2 - 10x + 9}\\) 7. \\(\frac{12x^2}{9x + 45} \cdot \frac{3x^3 - 75x}{36x^4}\\)

Explanation:

Problem 1: $\boldsymbol{\frac{9x^2 + 27x^3}{7x + 21x^2}}$

Step 1: Factor numerator and denominator

Numerator: $9x^2 + 27x^3 = 9x^2(1 + 3x)$
Denominator: $7x + 21x^2 = 7x(1 + 3x)$

Step 2: Cancel common factors

Cancel $(1 + 3x)$ and simplify:
$\frac{9x^2(1 + 3x)}{7x(1 + 3x)} = \frac{9x}{7}$ (for $x
eq 0, -\frac{1}{3}$)

Problem 2: $\boldsymbol{\frac{x^2 - 3x - 10}{x^2 - 4x - 12}}$

Step 1: Factor numerator and denominator

Numerator: $x^2 - 3x - 10 = (x - 5)(x + 2)$
Denominator: $x^2 - 4x - 12 = (x - 6)(x + 2)$

Step 2: Cancel common factors

Cancel $(x + 2)$ and simplify:
$\frac{(x - 5)(x + 2)}{(x - 6)(x + 2)} = \frac{x - 5}{x - 6}$ (for $x
eq -2, 6$)

Problem 3: $\boldsymbol{\frac{x^2 + 9x + 18}{36 - x^2}}$

Step 1: Factor numerator and denominator

Numerator: $x^2 + 9x + 18 = (x + 3)(x + 6)$
Denominator: $36 - x^2 = (6 - x)(6 + x) = -(x - 6)(x + 6)$ (difference of squares)

Step 2: Cancel common factors

Cancel $(x + 6)$ and simplify:
$\frac{(x + 3)(x + 6)}{-(x - 6)(x + 6)} = \frac{x + 3}{-(x - 6)} = \frac{-x - 3}{x - 6}$ (or $\frac{x + 3}{6 - x}$) (for $x
eq -6, 6$)

Problem 4: $\boldsymbol{\frac{2x - 2}{x} \cdot \frac{3x^2}{9x - 9}}$

Answer:

s:

1. $\boldsymbol{\frac{9x}{7}}$
2. $\boldsymbol{\frac{x - 5}{x - 6}}$
3. $\boldsymbol{\frac{-x - 3}{x - 6}}$ (or $\boldsymbol{\frac{x + 3}{6 - x}}$)
4. $\boldsymbol{\frac{2x}{3}}$
5. $\boldsymbol{\frac{x}{x - 4}}$
6. $\boldsymbol{\frac{x - 3}{x - 2}}$
7. $\boldsymbol{\frac{x - 5}{9x}}$