Absolutely, I'll guide you through rewriting the given rational expressions with a common denominator step-by-step.
### Step 1: Factor the Denominators
First, we need to factor the denominators of the given fractions.
Given:
[tex]\[ \frac{8}{5c^2 - 6c - 27} \][/tex]
[tex]\[ \frac{2c}{5c^2 - 31c - 72} \][/tex]
To factor the quadratic denominators, we find the factors for each.
#### For [tex]\( 5c^2 - 6c - 27 \)[/tex]:
Factor the quadratic expression:
[tex]\[ 5c^2 - 6c - 27 = (5c + 9)(c - 3) \][/tex]
Verification:
Expanding [tex]\( (5c + 9)(c - 3) \)[/tex]:
[tex]\[ 5c \cdot c - 5c \cdot 3 + 9 \cdot c + 9 \cdot -3 = 5c^2 - 15c + 9c - 27 = 5c^2 - 6c - 27 \][/tex]
So, verified as correct.
#### For [tex]\( 5c^2 - 31c - 72 \)[/tex]:
Factor the quadratic expression:
[tex]\[ 5c^2 - 31c - 72 = (5c + 9)(c - 8) \][/tex]
Verification:
Expanding [tex]\( (5c + 9)(c - 8) \)[/tex]:
[tex]\[ 5c \cdot c - 5c \cdot 8 + 9 \cdot c + 9 \cdot -8 = 5c^2 - 40c + 9c - 72 = 5c^2 - 31c - 72 \][/tex]
Again, verified as correct.
### Step 2: Determine the Common Denominator
The common denominator for both fractions is:
[tex]\[ (5c+9)(c-3)(c-8) \][/tex]
### Step 3: Rewrite the Fractions with the Common Denominator
To rewrite each fraction with the common denominator, we will adjust the numerators accordingly.
#### For the first fraction:
[tex]\[ \frac{8}{5c^2 - 6c - 27} \][/tex]
Using the factorization:
[tex]\[ 5c^2 - 6c - 27 = (5c + 9)(c - 3) \][/tex]
To have the common denominator [tex]\((5c+9)(c-3)(c-8)\)[/tex], we multiply the numerator and denominator by [tex]\((c - 8)\)[/tex]:
[tex]\[ \frac{8}{(5c + 9)(c - 3)} \cdot \frac{(c - 8)}{(c - 8)} = \frac{8(c - 8)}{(5c + 9)(c - 3)(c - 8)} = \frac{8c - 64}{(5c + 9)(c - 3)(c - 8)} \][/tex]
#### For the second fraction:
[tex]\[ \frac{2c}{5c^2 - 31c - 72} \][/tex]
Using the factorization:
[tex]\[ 5c^2 - 31c - 72 = (5c + 9)(c - 8) \][/tex]
To have the common denominator [tex]\((5c+9)(c-3)(c-8)\)[/tex], we multiply the numerator and denominator by [tex]\((c - 3)\)[/tex]:
[tex]\[ \frac{2c}{(5c + 9)(c - 8)} \cdot \frac{(c - 3)}{(c - 3)} = \frac{2c(c - 3)}{(5c + 9)(c - 8)(c - 3)} = \frac{2c^2 - 6c}{(5c + 9)(c - 8)(c - 3)} \][/tex]
### Conclusion
The equivalent rational expressions with the common denominator [tex]\((5c+9)(c-3)(c-8)\)[/tex] are:
[tex]\[ \frac{8c - 64}{(5c + 9)(c - 3)(c - 8)} \][/tex]
[tex]\[ \frac{2c^2 - 6c}{(5c + 9)(c - 8)(c - 3)} \][/tex]
These are the rewritten forms of the given fractions with the common denominator.
