Question

Find two rational expressions a / b and c / d that produce the result x − 1 / x2 when using the following operations. Answers
for each type of operation may vary. Justify your answers.
a. Addition
b. Subtraction
c. Multiplication
d. Division

Answer 1

Answer:

a) Let $\frac{a}{b}=\frac{-1}{x^2}, \text{ and } \frac{c}{d}=\frac{1}{x}$.

Observe that

$\frac{a}{b}+\frac{c}{d}=\frac{-1}{x^2}+\frac{1}{x}=\frac{-x+x^2}{x^3}=\frac{x(x-1)}{xx^2}=\frac{x-1}{x^2}$

b)

Let $\frac{a}{b}=\frac{1}{x}, \text{ and } \frac{c}{d}=\frac{1}{x^2}.$

Observe that

$\frac{a}{b}-\frac{c}{d}=\frac{1}{x}-\frac{1}{x^2}=\frac{x^2-x}{x^3}=\frac{x(x-1)}{xx^2}=\frac{x-1}{x^2}$

c)

Let $\frac{a}{b}=\frac{x-1}{x}, \text{ and } \frac{c}{d}=\frac{1}{x}.$

Observe that

$\frac{a}{b}*\frac{c}{d}=\frac{x-1}{x}*\frac{1}{x}=\frac{(x-1)1}{x*x}=\frac{x-1}{x^2}$

d)

Let $\frac{a}{b}=\frac{x-1}{x}, \text{ and } \frac{c}{d}=\frac{x}{1}.$

Observe that

$\frac{a}{b}\div\frac{c}{d}=\frac{x-1}{x}\div\frac{x}{1}=\frac{x-1}{x}*\frac{1}{x}=\frac{x-1}{x^2}$