Sure, let's work through rewriting the given rational expressions with a common denominator, which is [tex]\((5z - 4)(z - 3)(z + 5)\)[/tex].
### Step 1: Factor Denominators
1. Factor the denominator of [tex]\(\frac{5}{5z^2 - 19z + 12}\)[/tex]:
The expression [tex]\(5z^2 - 19z + 12\)[/tex] can be factored as:
[tex]\[
5z^2 - 19z + 12 = (z - 3)(5z - 4)
\][/tex]
2. Factor the denominator of [tex]\(\frac{10z}{5z^2 + 21z - 20}\)[/tex]:
The expression [tex]\(5z^2 + 21z - 20\)[/tex] can be factored as:
[tex]\[
5z^2 + 21z - 20 = (z + 5)(5z - 4)
\][/tex]
### Step 2: Rewriting Each Expression with the Common Denominator
To rewrite each expression with the common denominator [tex]\((5z - 4)(z - 3)(z + 5)\)[/tex], we need to adjust the numerators accordingly.
1. Rewriting [tex]\(\frac{5}{(z - 3)(5z - 4)}\)[/tex]:
The denominator [tex]\((z - 3)(5z - 4)\)[/tex] is missing the factor [tex]\((z + 5)\)[/tex] to match the common denominator [tex]\((5z - 4)(z - 3)(z + 5)\)[/tex].
Multiply both the numerator and the denominator by [tex]\((z + 5)\)[/tex]:
[tex]\[
\frac{5}{(z - 3)(5z - 4)} = \frac{5(z + 5)}{(z - 3)(5z - 4)(z + 5)}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{5(z + 5)}{(5z - 4)(z - 3)(z + 5)} = \frac{5z + 25}{(5z - 4)(z - 3)(z + 5)}
\][/tex]
2. Rewriting [tex]\(\frac{10z}{(z + 5)(5z - 4)}\)[/tex]:
The denominator [tex]\((z + 5)(5z - 4)\)[/tex] is missing the factor [tex]\((z - 3)\)[/tex].
Multiply both the numerator and the denominator by [tex]\((z - 3)\)[/tex]:
[tex]\[
\frac{10z}{(z + 5)(5z - 4)} = \frac{10z(z - 3)}{(z + 5)(5z - 4)(z - 3)}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{10z(z - 3)}{(z + 5)(5z - 4)(z - 3)} = \frac{10z^2 - 30z}{(5z - 4)(z - 3)(z + 5)}
\][/tex]
### Final Answer:
The equivalent rational expressions with the common denominator [tex]\((5z - 4)(z - 3)(z + 5)\)[/tex] are:
[tex]\[
\frac{5}{5z^2 - 19z + 12} = \frac{5z + 25}{(5z - 4)(z - 3)(z + 5)}
\][/tex]
[tex]\[
\frac{10z}{5z^2 + 21z - 20} = \frac{10z^2 - 30z}{(5z - 4)(z - 3)(z + 5)}
\][/tex]
