Question

Please help me with this with full solutions!!!

Answer 1

Option (1)

Explanation:

The given expression is,

`(x^(2)-19x+90)/(x^(2)+19x+90)` ÷
`(x^(2)-2x-80)/(x^(2)-x-72)`

Now we will factor each polynomial separately and substitute the factors in the expression.

x² - 19x + 90 = x² - 10x - 9x + 90

= x(x - 10) - 9(x - 10)

= (x - 10)(x - 9)

x² - 2x - 80 = x² - 10x + 8x - 80

= x(x - 10) + 8(x - 10)

= (x + 8)(x - 10)

x²- x - 72 = x² - 9x + 8x - 72

= x(x - 9) + 8(x - 9)

= (x + 8)(x - 9)

x² + 19x + 90 = x² + 10x + 9x + 90

= x(x + 10) + 9(x + 10)

= (x + 9)(x + 10)

Now by substituting these factors in the given expression

`((x - 10)(x - 9))/((x + 10)(x + 9))` ÷
`((x + 8)(x - 10))/((x + 8)(x - 9))` =
`((x - 10)(x - 9))/((x + 10)(x + 9))` ÷
`((x - 10))/((x - 9))`

=
`((x - 10)(x - 9))/((x + 10)(x + 9))` ×
`((x - 9))/((x - 10))`

=
`((x - 9)^2)/((x + 10)(x + 9))`

Therefore, Option (1) will be the answer.