Calculus Examples

$\frac{z + 3}{z^{2} + 3 z + 2} - \frac{z - 1}{z^{2} + z - 2}$

Step 1

Simplify each term.

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Step 1.1

Factor $z^{2} + 3 z + 2$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Factor $z^{2} + z - 2$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{z - 1}{(z - 1) (z + 2)}$

Step 1.3

Cancel the common factor of $z - 1$ .

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Step 1.3.1

Cancel the common factor.

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{z - 1}{(z - 1) (z + 2)}$

Step 1.3.2

Rewrite the expression.

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{1}{z + 2}$

Step 2

To write $- \frac{1}{z + 2}$ as a fraction with a common denominator, multiply by $\frac{z + 1}{z + 1}$ .

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{1}{z + 2} \cdot \frac{z + 1}{z + 1}$

Step 3

Write each expression with a common denominator of $(z + 1) (z + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{z + 2}$ by $\frac{z + 1}{z + 1}$ .

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{z + 1}{(z + 2) (z + 1)}$

Step 3.2

Reorder the factors of $(z + 2) (z + 1)$ .

$\frac{z + 3}{(z + 1) (z + 2)} - \frac{z + 1}{(z + 1) (z + 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{z + 3 - (z + 1)}{(z + 1) (z + 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{z + 3 - z - 1 \cdot 1}{(z + 1) (z + 2)}$

Step 5.2

Multiply $- 1$ by $1$ .

$\frac{z + 3 - z - 1}{(z + 1) (z + 2)}$

Step 5.3

Subtract $z$ from $z$ .

$\frac{0 + 3 - 1}{(z + 1) (z + 2)}$

Step 5.4

Add $0$ and $3$ .

$\frac{3 - 1}{(z + 1) (z + 2)}$

Step 5.5

Subtract $1$ from $3$ .

$\frac{2}{(z + 1) (z + 2)}$