Simplify (2x+5)/(x^3)-(x+2)/(x^2)

$\frac{2 x + 5}{x^{3}} - \frac{x + 2}{x^{2}}$

Step 1

To write $- \frac{x + 2}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x + 5}{x^{3}} - \frac{x + 2}{x^{2}} \cdot \frac{x}{x}$

Step 2

Write each expression with a common denominator of $x^{3}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 2}{x^{2}}$ by $\frac{x}{x}$ .

$\frac{2 x + 5}{x^{3}} - \frac{(x + 2) x}{x^{2} x}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{2 x + 5}{x^{3}} - \frac{(x + 2) x}{x^{2} x^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x + 5}{x^{3}} - \frac{(x + 2) x}{x^{2 + 1}}$

Add $2$ and $1$ .

$\frac{2 x + 5}{x^{3}} - \frac{(x + 2) x}{x^{3}}$

Step 3

Combine the numerators over the common denominator.

$\frac{2 x + 5 - (x + 2) x}{x^{3}}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{2 x + 5 + (- x - 1 \cdot 2) x}{x^{3}}$

Multiply $- 1$ by $2$ .

$\frac{2 x + 5 + (- x - 2) x}{x^{3}}$

Apply the distributive property.

$\frac{2 x + 5 - x \cdot x - 2 x}{x^{3}}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 x + 5 - (x \cdot x) - 2 x}{x^{3}}$

Multiply $x$ by $x$ .

$\frac{2 x + 5 - x^{2} - 2 x}{x^{3}}$

Subtract $2 x$ from $2 x$ .

$\frac{5 - x^{2} + 0}{x^{3}}$

Add $5 - x^{2}$ and $0$ .

$\frac{5 - x^{2}}{x^{3}}$