Answer:
[tex]\displaystyle \mathbf{\frac{5}{(c+4)}}[/tex]
Step-by-step explanation:
Division of Rational Expressions
When dividing two rational expressions like f(c) / g(c), it's usually easier to multiply the numerator by the reciprocal of the denominator: f(c) * (1/g(c)). The reciprocal is just flipping the numerator and the denominator.
Let's find the following division:
[tex]\displaystyle \frac{\frac{c^2-64}{3c^2+26c+16}} {\frac{c^2-4c-32}{15c+10}}[/tex]
First, we factor the polynomials where possible.
[tex]c^2-64 = (c-8)(c+8)[/tex]
[tex]3c^2+26c+16=(3c+2)(c+8)[/tex]
[tex]c^2-4c-32=(c-8)(c+4)[/tex]
[tex]15c+10=5(3c+2)[/tex]
Substituting:
[tex]\displaystyle \frac{\frac{c^2-64}{3c^2+26c+16}} {\frac{c^2-4c-32}{15c+10}}=\frac{\frac{(c-8)(c+8)}{(3c+2)(c+8)}} {\frac{(c-8)(c+4)}{5(3c+2)}}[/tex]
Multiplying by the reciprocal of the denominator:
[tex]\displaystyle \frac{\frac{c^2-64}{3c^2+26c+16}} {\frac{c^2-4c-32}{15c+10}}=\frac{(c-8)(c+8)}{(3c+2)(c+8)}\cdot \frac{5(3c+2)}{(c-8)(c+4)}[/tex]
Simplifying by (c+8)(3c+2)(c-8):
[tex]\displaystyle \frac{\frac{c^2-64}{3c^2+26c+16}} {\frac{c^2-4c-32}{15c+10}}=\frac{5}{(c+4)}[/tex]
Answer:
[tex]\displaystyle \mathbf{\mathbf{\frac{5}{(c+4)}}}[/tex]
