Rewrite the rational expression as an equivalent rational expression with the given denominator. (Both are fractions)
It's 21 over x squared minus 6x is equal to 1 over x (x-6)(x+5)
-21 = 1
x2 -6x x(x-6)(x+5)
To rewrite the rational expression (21 / (x^2 - 6x)) as an equivalent rational expression with the denominator (x(x-6)(x+5)), we need to multiply the numerator and denominator by the appropriate factors in order to achieve the desired denominator.
Let's break down the steps:
1. Start with the given expression: 21 / (x^2 - 6x).
2. Factor the denominator as much as possible: x^2 - 6x = x(x - 6).
3. Determine the missing factors in the desired denominator (x(x-6)(x+5)) that are not already present in the current denominator (x(x - 6)). Here, the missing factor is (x+5).
4. Multiply the numerator and denominator by the missing factor (x+5) to introduce it into the expression:
(21 / (x^2 - 6x)) * ((x+5) / (x+5)).
This step ensures that we are not changing the value of the expression, as multiplying by 1 (in the form of (x+5) / (x+5)) does not alter the value.
5. Simplify the expression by performing the multiplication:
21(x + 5) / (x(x - 6)(x + 5)).
6. Simplify further if desired:
(21x + 105) / (x(x - 6)(x + 5)).
Thus, the given rational expression (21 / (x^2 - 6x)) can be rewritten as (21x + 105) / (x(x - 6)(x + 5)).