To rewrite the given rational expressions with a common denominator [tex]\((4y + 5)(y - 7)(y + 8)\)[/tex], follow these steps:
### Given Rational Expressions
1. [tex]\(\frac{2y}{4y^2 - 23y - 35}\)[/tex]
2. [tex]\(\frac{4}{4y^2 + 37y + 40}\)[/tex]
### Step 1: Factor the Denominators
First, factor the denominators of each expression.
Denominator of the first expression:
[tex]\[ 4y^2 - 23y - 35 \][/tex]
This factors into:
[tex]\[ (y - 7)(4y + 5) \][/tex]
Denominator of the second expression:
[tex]\[ 4y^2 + 37y + 40 \][/tex]
This factors into:
[tex]\[ (y + 8)(4y + 5) \][/tex]
### Step 2: Establish the Common Denominator
The common denominator is given as:
[tex]\[ (4y + 5)(y - 7)(y + 8) \][/tex]
### Step 3: Rewrite Each Expression
To express each rational expression with the common denominator, we need to adjust the numerator accordingly so that the entire expression remains equivalent to the original.
First Expression:
Given:
[tex]\[ \frac{2y}{(y - 7)(4y + 5)} \][/tex]
To change the denominator to the common denominator [tex]\((4y + 5)(y - 7)(y + 8)\)[/tex], multiply the numerator and the denominator by the missing factor [tex]\((y + 8)\)[/tex]:
[tex]\[ \frac{2y \cdot (y + 8)}{(y - 7)(4y + 5) \cdot (y + 8)} = \frac{2y(y + 8)}{(4y + 5)(y - 7)(y + 8)} \][/tex]
So, the numerator for the first expression in terms of the common denominator is:
[tex]\[ 2y(y + 8) \][/tex]
Second Expression:
Given:
[tex]\[ \frac{4}{(y + 8)(4y + 5)} \][/tex]
To change the denominator to the common denominator [tex]\((4y + 5)(y - 7)(y + 8)\)[/tex], multiply the numerator and the denominator by the missing factor [tex]\((y - 7)\)[/tex]:
[tex]\[ \frac{4 \cdot (y - 7)}{(y + 8)(4y + 5) \cdot (y - 7)} = \frac{4(y - 7)}{(4y + 5)(y - 7)(y + 8)} \][/tex]
So, the numerator for the second expression in terms of the common denominator is:
[tex]\[ 4(y - 7) \][/tex]
### Final Equivalent Rational Expressions
Therefore, the given rational expressions with the common denominator [tex]\((4y + 5)(y - 7)(y + 8)\)[/tex] are:
1. [tex]\(\frac{2y(y + 8)}{(4y + 5)(y - 7)(y + 8)}\)[/tex]
2. [tex]\(\frac{4(y - 7)}{(4y + 5)(y - 7)(y + 8)}\)[/tex]
Thus, we have successfully rewritten the expressions with the common denominator.
