(x - 1)/x² - x - 20, give the equivalent numerator if the denominator is (x - 5)(x + 4)(x + 3).

Solution:

Given, the expression is (x - 1)/x² - x - 20

We have to find the equivalent numerator if the denominator is (x - 5)(x + 4)(x + 3).

The factors of x² - x - 20 = x² - 5x - 4x - 20

= x(x - 5) + 4(x - 5)

= (x + 4)(x - 5)

So,  (x - 1)/x² - x - 20 =  (x - 1)/[(x + 4)(x - 5)]

Now,   (x - 1) / [(x + 4)(x - 5)] = numerator / [(x + 4)(x + 3)(x - 5)]

On rearranging and multiplying with (x + 3) in the RHS with Numerator and Denominator,

Numerator/[(x + 4)(x + 3)(x - 5)] = (x - 1)(x + 3)/[(x + 4)(x + 3)(x - 5)]

So, numerator = (x - 1)(x + 3)

Therefore, the numerator is (x - 1)(x + 3).

(x - 1)/x² - x - 20, give the equivalent numerator if the denominator is (x - 5)(x + 4)(x + 3).

Summary:

The equivalent numerator of (x - 1)/x² - x - 20 if the denominator is (x - 5)(x + 4)(x + 3) is (x - 1)(x + 3).