21 OVER X squared MINUS 6x is equal to 1 OVER x(x-6)(x+5).
I have to rewrite the ration expression as an equivalent rational expression with the given denominator.
x^2 - 6 x = x(x-6)
So, assuming that x is not equal to zero or 6, multiply both sides by
x (x-6):
21 = 1/(x+5) ----->
x = 1/21 - 5 = - 104/21
To rewrite the rational expression 21/(x^2 - 6x) as an equivalent expression with the denominator x(x-6)(x+5), we need to find the missing factors in the denominator.
First, factor out the expression x^2 - 6x:
x^2 - 6x = x(x-6)
Notice that the denominator, x(x-6)(x+5), already contains the factors x and (x-6).
Now, we need to introduce the missing factor of (x+5) in the numerator. To do this, we multiply the numerator and denominator by (x+5):
(21/(x^2 - 6x)) * ((x+5)/(x+5))
Simplifying the numerator and denominator:
(21 * (x+5)) / (x(x-6)(x+5))
This gives us the equivalent rational expression with the desired denominator:
21(x+5) / (x(x-6)(x+5))
Simplifying further, we can cancel out the common factor of (x+5):
21 / (x(x-6))
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 decompose the given denominator into its individual factors.
The given denominator x(x - 6)(x + 5) is a product of three factors: x, (x - 6), and (x + 5). We can set up three separate fractions with these factors as their denominators:
1/x, 1/(x - 6), and 1/(x + 5).
Now, we can rewrite the original rational expression using these decomposed denominators:
21/(x^2 - 6x) = 21/(x(x - 6)).
To combine this expression with the decomposed denominators, we need to multiply the numerator and denominator by the missing factors. In this case, we need to multiply by (x - 6) and (x + 5) to get the common denominator:
21/(x(x - 6)) = (21/(x(x - 6))) * ((x - 6)(x + 5)/(x - 6)(x + 5)).
Now, we can simplify the expression:
(21/(x(x - 6))) * ((x - 6)(x + 5)/(x - 6)(x + 5)) = (21(x - 6)(x + 5))/((x(x - 6))(x - 6)(x + 5)).
Finally, we can cancel the common factors to obtain the equivalent rational expression:
(21(x - 6)(x + 5))/((x(x - 6))(x - 6)(x + 5)) = 21/x(x - 6).
Therefore, the equivalent rational expression with the denominator x(x - 6)(x + 5) is 21/x(x - 6).