Solve 5+p^2/p^2-36-6p+p^2/36p+p^2

$$\frac { 5 + p ^ { 2 } } { p ^ { 2 } - 36 } - \frac { 6 p + p ^ { 2 } } { 36 p + p ^ { 2 } }$$

Evaluate

$$\frac{30p^{2}+41p+396}{\left(p+36\right)\left(p^{2}-36\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{6p+p^{2}}{36p+p^{2}}.

$$\frac{5+p^{2}}{p^{2}-36}-\frac{p\left(p+6\right)}{p\left(p+36\right)}$$

Cancel out p in both numerator and denominator.

$$\frac{5+p^{2}}{p^{2}-36}-\frac{p+6}{p+36}$$

Factor p^{2}-36.

$$\frac{5+p^{2}}{\left(p-6\right)\left(p+6\right)}-\frac{p+6}{p+36}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(p-6\right)\left(p+6\right) and p+36 is \left(p-6\right)\left(p+6\right)\left(p+36\right). Multiply \frac{5+p^{2}}{\left(p-6\right)\left(p+6\right)} times \frac{p+36}{p+36}. Multiply \frac{p+6}{p+36} times \frac{\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)}.

$$\frac{\left(5+p^{2}\right)\left(p+36\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}-\frac{\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Since \frac{\left(5+p^{2}\right)\left(p+36\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)} and \frac{\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5+p^{2}\right)\left(p+36\right)-\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Do the multiplications in \left(5+p^{2}\right)\left(p+36\right)-\left(p+6\right)\left(p-6\right)\left(p+6\right).

$$\frac{5p+180+p^{3}+36p^{2}-p^{3}+36p-6p^{2}+216}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Combine like terms in 5p+180+p^{3}+36p^{2}-p^{3}+36p-6p^{2}+216.

$$\frac{41p+396+30p^{2}}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Expand \left(p-6\right)\left(p+6\right)\left(p+36\right).

$$\frac{41p+396+30p^{2}}{p^{3}+36p^{2}-36p-1296}$$

Expand

$$\frac{30p^{2}+41p+396}{\left(p+36\right)\left(p^{2}-36\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{6p+p^{2}}{36p+p^{2}}.

$$\frac{5+p^{2}}{p^{2}-36}-\frac{p\left(p+6\right)}{p\left(p+36\right)}$$

Cancel out p in both numerator and denominator.

$$\frac{5+p^{2}}{p^{2}-36}-\frac{p+6}{p+36}$$

Factor p^{2}-36.

$$\frac{5+p^{2}}{\left(p-6\right)\left(p+6\right)}-\frac{p+6}{p+36}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(p-6\right)\left(p+6\right) and p+36 is \left(p-6\right)\left(p+6\right)\left(p+36\right). Multiply \frac{5+p^{2}}{\left(p-6\right)\left(p+6\right)} times \frac{p+36}{p+36}. Multiply \frac{p+6}{p+36} times \frac{\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)}.

$$\frac{\left(5+p^{2}\right)\left(p+36\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}-\frac{\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Since \frac{\left(5+p^{2}\right)\left(p+36\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)} and \frac{\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5+p^{2}\right)\left(p+36\right)-\left(p+6\right)\left(p-6\right)\left(p+6\right)}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Do the multiplications in \left(5+p^{2}\right)\left(p+36\right)-\left(p+6\right)\left(p-6\right)\left(p+6\right).

$$\frac{5p+180+p^{3}+36p^{2}-p^{3}+36p-6p^{2}+216}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Combine like terms in 5p+180+p^{3}+36p^{2}-p^{3}+36p-6p^{2}+216.

$$\frac{41p+396+30p^{2}}{\left(p-6\right)\left(p+6\right)\left(p+36\right)}$$

Expand \left(p-6\right)\left(p+6\right)\left(p+36\right).

$$\frac{41p+396+30p^{2}}{p^{3}+36p^{2}-36p-1296}$$