Question

find the lcd for each set of rational expressions.

1. $\frac{x}{2x + 16}$ and $\frac{-4x}{3x - 27}$
2. $\frac{x^2 - 4}{5x - 30}$ and $\frac{5x + 13}{7x - 42}$
3. $\frac{4x + 12}{x^2 + 5x + 6}$ and $\frac{5x + 15}{10x + 20}$
4. $\frac{-11}{x^2 - 3x - 28}$ and $\frac{2}{x^2 - 2x - 24}$
5. $\frac{12}{3x^2 - 21x - 54}$ and $\frac{-1}{21x^2 - 84}$
6. $\frac{3x}{5x^2 - 40x + 60}$ and $\frac{17}{-7x^2 + 56x - 84}$

Explanation:

Problem 9

Step1: Factor denominators

$2x+16=2(x+8)$, $3x-27=3(x-9)$

Step2: Identify unique factors

Unique factors: $2, 3, (x+8), (x-9)$

Step3: Multiply unique factors

$\text{LCD}=2\times3\times(x+8)(x-9)=6(x+8)(x-9)$

Problem 10

Step1: Factor denominators

$5x-30=5(x-6)$, $7x-42=7(x-6)$; factor numerator $x^2-4=(x-2)(x+2)$ (not needed for LCD)

Step2: Identify unique factors

Unique factors: $5, 7, (x-6)$

Step3: Multiply unique factors

$\text{LCD}=5\times7\times(x-6)=35(x-6)$

Problem 11

Step1: Factor all parts

$4x+12=4(x+3)$, $x^2+5x+6=(x+2)(x+3)$; $5x+15=5(x+3)$, $10x+20=10(x+2)$

Step2: Identify denominators' factors

Denominators: $(x+2)(x+3)$, $10(x+2)$

Step3: Multiply unique factors

$\text{LCD}=10(x+2)(x+3)$

Problem 12

Step1: Factor denominators

$x^2-3x-28=(x-7)(x+4)$, $x^2-2x-24=(x-6)(x+4)$

Step2: Identify unique factors

Unique factors: $(x-7), (x+4), (x-6)$

Step3: Multiply unique factors

$\text{LCD}=(x-7)(x+4)(x-6)$

Problem 13

Step1: Factor denominators

$3x^2-21x-54=3(x^2-7x-18)=3(x-9)(x+2)$; $21x^2-84=21(x^2-4)=21(x-2)(x+2)$

Step2: Identify unique factors

Unique factors: $3, 7, (x-9), (x+2), (x-2)$

Step3: Multiply unique factors

$\text{LCD}=21(x-9)(x+2)(x-2)$

Problem 14

Step1: Factor denominators

$5x^2-40x+60=5(x^2-8x+12)=5(x-2)(x-6)$; $-7x^2+56x-84=-7(x^2-8x+12)=-7(x-2)(x-6)$

Step2: Identify unique factors

Unique factors: $5, 7, (x-2), (x-6)$

Step3: Multiply unique factors

$\text{LCD}=35(x-2)(x-6)$

Answer:

1. $6(x+8)(x-9)$
2. $35(x-6)$
3. $10(x+2)(x+3)$
4. $(x-7)(x+4)(x-6)$
5. $21(x-9)(x+2)(x-2)$
6. $35(x-2)(x-6)$