Let's solve the given problem step-by-step.
We need to rewrite the rational expressions [tex]\(\frac{4}{4y^2 + 23y - 35}\)[/tex] and [tex]\(\frac{2y}{4y^2 - 37y + 40}\)[/tex] with the common denominator [tex]\((4y - 5)(y + 7)(y - 8)\)[/tex].
### Step 1: Factor the denominators
First, factor the denominators of the given fractions.
1. For the first fraction [tex]\(\frac{4}{4y^2 + 23y - 35}\)[/tex]:
[tex]\[
4y^2 + 23y - 35
\][/tex]
We need to find factors of [tex]\(4y^2 + 23y - 35\)[/tex].
We use the factoring method to get:
[tex]\[
(4y^2 + 23y - 35) = (4y - 5)(y + 7)
\][/tex]
2. For the second fraction [tex]\(\frac{2y}{4y^2 - 37y + 40}\)[/tex]:
[tex]\[
4y^2 - 37y + 40
\][/tex]
Likewise, we factorize [tex]\(4y^2 - 37y + 40\)[/tex] and get:
[tex]\[
(4y^2 - 37y + 40) = (4y - 5)(y - 8)
\][/tex]
### Step 2: Determine the least common denominator (LCD)
The least common denominator (LCD) for both fractions is the product of their unique factors. Given, the common denominator:
[tex]\[
(4y - 5)(y + 7)(y - 8)
\][/tex]
### Step 3: Rewrite each fraction with the common denominator
Now, we rewrite each fraction with this common denominator.
1. Rewrite [tex]\(\frac{4}{(4y - 5)(y + 7)}\)[/tex] with the common denominator [tex]\((4y - 5)(y + 7)(y - 8)\)[/tex]:
[tex]\[
\frac{4}{(4y - 5)(y + 7)}
\][/tex]
To have the common denominator, multiply the numerator and denominator by the missing term [tex]\((y - 8)\)[/tex]:
[tex]\[
\frac{4 \cdot (y - 8)}{(4y - 5)(y + 7)(y - 8)} = \frac{4(y - 8)}{(4y - 5)(y + 7)(y - 8)}
\][/tex]
[tex]\[
\frac{4(y - 8)}{(4y - 5)(y + 7)(y - 8)} = \frac{4y - 32}{(4y - 5)(y + 7)(y - 8)}
\][/tex]
2. Rewrite [tex]\(\frac{2y}{(4y - 5)(y - 8)}\)[/tex] with the common denominator [tex]\((4y - 5)(y + 7)(y - 8)\)[/tex]:
[tex]\[
\frac{2y}{(4y - 5)(y - 8)}
\][/tex]
To have the common denominator, multiply the numerator and denominator by the missing term [tex]\((y + 7)\)[/tex]:
[tex]\[
\frac{2y \cdot (y + 7)}{(4y - 5)(y - 8)(y + 7)} = \frac{2y(y + 7)}{(4y - 5)(y - 8)(y + 7)}
\][/tex]
[tex]\[
\frac{2y(y + 7)}{(4y - 5)(y - 8)(y + 7)} = \frac{2y^2 + 14y}{(4y - 5)(y - 8)(y + 7)}
\][/tex]
### Completed Equivalent Rational Expressions
The equivalent rational expressions with the common denominator [tex]\((4y - 5)(y + 7)(y - 8)\)[/tex] are:
1. [tex]\(\frac{4y - 32}{(4y - 5)(y + 7)(y - 8)}\)[/tex]
2. [tex]\(\frac{2y^2 + 14y}{(4y - 5)(y - 8)(y + 7)}\)[/tex]
