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

a) Let
`(a)/(b)=(-1)/(x^2), \text{ and } (c)/(d)=(1)/(x)`.

Observe that

`(a)/(b)+(c)/(d)=(-1)/(x^2)+(1)/(x)=(-x+x^2)/(x^3)=(x(x-1))/(xx^2)=(x-1)/(x^2)`

b)

Let
`(a)/(b)=(1)/(x), \text{ and } (c)/(d)=(1)/(x^2).`

Observe that

`(a)/(b)-(c)/(d)=(1)/(x)-(1)/(x^2)=(x^2-x)/(x^3)=(x(x-1))/(xx^2)=(x-1)/(x^2)`

c)

Let
`(a)/(b)=(x-1)/(x), \text{ and } (c)/(d)=(1)/(x).`

Observe that

`(a)/(b)*(c)/(d)=(x-1)/(x)*(1)/(x)=((x-1)1)/(x*x)=(x-1)/(x^2)`

d)

Let
`(a)/(b)=(x-1)/(x), \text{ and } (c)/(d)=(x)/(1).`

Observe that

`(a)/(b)/(c)/(d)=(x-1)/(x)/(x)/(1)=(x-1)/(x)*(1)/(x)=(x-1)/(x^2)`