Question

Please help me with this with full solutions!!!

Answer 1

Answer:

Option (1)

Step-by-step explanation:

The given expression is,

$\frac{x^{2}-19x+90}{x^{2}+19x+90}$ ÷ $\frac{x^{2}-2x-80}{x^{2}-x-72}$

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

$\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}$ ÷ $\frac{(x + 8)(x - 10)}{(x + 8)(x - 9)}$ = $\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}$ ÷ $\frac{(x - 10)}{(x - 9)}$

                                     = $\frac{(x - 10)(x - 9)}{(x + 10)(x + 9)}$ × $\frac{(x - 9)}{(x - 10)}$

                                     = $\frac{(x - 9)^2}{(x + 10)(x + 9)}$

Therefore, Option (1) will be the answer.