SOLUTION: Write each rational expression as an equivalent expression with the indicated denominator. \frac{14}{z^2-3z}=\frac{?}{z\left(z-3\right)\left(z-2\right)}

Algebra ->  Equations -> SOLUTION: Write each rational expression as an equivalent expression with the indicated denominator. \frac{14}{z^2-3z}=\frac{?}{z\left(z-3\right)\left(z-2\right)}      Log On


   

Question 1201042: Write each rational expression as an equivalent expression with the indicated denominator.
\frac{14}{z^2-3z}=\frac{?}{z\left(z-3\right)\left(z-2\right)}
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

LHS = left hand side
RHS = right hand side

Let n be the unknown numerator expression in the RHS fraction.
14%2F%28z%5E2-3z%29+=+n%2F%28z%28z-3%29%28z-2%29%29
n is some expression in terms of z.

Let's factor the denominator of the LHS.
z%5E2-3z+=+z%28z-3%29

This means
14%2F%28z%5E2-3z%29+=+n%2F%28z%28z-3%29%28z-2%29%29
is the same as
14%2F%28z%28z-3%29%29+=+n%2F%28z%28z-3%29%28z-2%29%29

The factors for the LHS denominator are z and (z-3). Both of which are present in the RHS denominator z%28z-3%29%28z-2%29. The LHS denominator is missing the factor (z-2).

We'll multiply top and bottom of the LHS fraction by (z-2) to fill in that missing gap.

14%2F%28z%28z-3%29%29+=+n%2F%28z%28z-3%29%28z-2%29%29



%2814%2A%28z-2%29%29%2F%28z%28z-3%29%2A%28z-2%29%29+=+n%2F%28z%28z-3%29%28z-2%29%29

%2814z-14%2A2%29%2F%28z%28z-3%29%28z-2%29%29+=+n%2F%28z%28z-3%29%28z-2%29%29

%2814z-28%29%2F%28z%28z-3%29%28z-2%29%29+=+n%2F%28z%28z-3%29%28z-2%29%29

Both denominators of the LHS and RHS have the same exact factorization. The fractions are only equal if the numerators were the same. Therefore, we must have n = 14(z-2) = 14z-28

In other words,
14%2F%28z%5E2-3z%29+=+%2814z-28%29%2F%28z%28z-3%29%28z-2%29%29
after multiplying top and bottom by (z-2).