Simplify (8p^3+1)/(4p^3+20p^2-p-5)

$\frac{8 p^{3} + 1}{4 p^{3} + 20 p^{2} - p - 5}$

Step 1

Simplify the numerator.

Rewrite $8 p^{3}$ as ${(2 p)}^{3}$ .

$\frac{{(2 p)}^{3} + 1}{4 p^{3} + 20 p^{2} - p - 5}$

Rewrite $1$ as $1^{3}$ .

$\frac{{(2 p)}^{3} + 1^{3}}{4 p^{3} + 20 p^{2} - p - 5}$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 2 p$ and $b = 1$ .

$\frac{(2 p + 1) ({(2 p)}^{2} - (2 p) \cdot 1 + 1^{2})}{4 p^{3} + 20 p^{2} - p - 5}$

Simplify.

Apply the product rule to $2 p$ .

$\frac{(2 p + 1) (2^{2} p^{2} - (2 p) \cdot 1 + 1^{2})}{4 p^{3} + 20 p^{2} - p - 5}$

Raise $2$ to the power of $2$ .

$\frac{(2 p + 1) (4 p^{2} - (2 p) \cdot 1 + 1^{2})}{4 p^{3} + 20 p^{2} - p - 5}$

Multiply $2$ by $- 1$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p \cdot 1 + 1^{2})}{4 p^{3} + 20 p^{2} - p - 5}$

Multiply $- 2$ by $1$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1^{2})}{4 p^{3} + 20 p^{2} - p - 5}$

One to any power is one.

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{4 p^{3} + 20 p^{2} - p - 5}$

Step 2

Simplify the denominator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(4 p^{3} + 20 p^{2}) - p - 5}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{4 p^{2} (p + 5) - (p + 5)}$

Factor the polynomial by factoring out the greatest common factor, $p + 5$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(p + 5) (4 p^{2} - 1)}$

Rewrite $4 p^{2}$ as ${(2 p)}^{2}$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(p + 5) ({(2 p)}^{2} - 1)}$

Rewrite $1$ as $1^{2}$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(p + 5) ({(2 p)}^{2} - 1^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 p$ and $b = 1$ .

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(p + 5) (2 p + 1) (2 p - 1)}$

Step 3

Cancel the common factor of $2 p + 1$ .

Cancel the common factor.

$\frac{(2 p + 1) (4 p^{2} - 2 p + 1)}{(p + 5) (2 p + 1) (2 p - 1)}$

Rewrite the expression.

$\frac{4 p^{2} - 2 p + 1}{(p + 5) (2 p - 1)}$

Simplify (8p^3+1)/(4p^3+20p^2-p-5)

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