Divide Rational Expressions
In the following exercises, divide.

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]

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