Question

\frac{x-5}{x^4-2x^3}\cdot \frac{x^2-4}{x^2-3x-10} x 4 −2x 3 x−5 ​ ⋅ x 2 −3x−10 x 2 −4 ​

Answer 1

`(x-5)/(x^4-2x^3)\cdot (x^2-4)/(x^2-3x-10) = (1)/(x^3)`

Explanation:

Given

`(x-5)/(x^4-2x^3)\cdot (x^2-4)/(x^2-3x-10)`

Required

Solve

Express
`x^2-4` as difference of two squares

`(x-5)/(x^4-2x^3)\cdot ((x-2)(x+2))/(x^2-3x-10)`

Factorize
`x^4 - 2x^3`

`(x-5)/(x^3(x-2))\cdot ((x-2)(x+2))/(x^2-3x-10)`

Cancel out x - 2

`(x-5)/(x^3)\cdot (x+2)/(x^2-3x-10)`

Expand
`x^2 - 3x - 10`

`(x-5)/(x^3)\cdot (x+2)/(x^2+2x-5x-10)`

Factorize:

`(x-5)/(x^3)\cdot (x+2)/(x(x+2)-5(x+2))`

Factor out x + 2

`(x-5)/(x^3)\cdot (x+2)/((x-5)(x+2))`

Cancel out x - 5 and x + 2

`(1)/(x^3)\cdot (1)/(1)`

`(1)/(x^3)\cdot 1`

`(1)/(x^3)`

Hence:

`(x-5)/(x^4-2x^3)\cdot (x^2-4)/(x^2-3x-10) = (1)/(x^3)`