Simplify (z^2+11z+30)/(z^2+z-20)-(z^2+2z-24)/(z^2+9z+18)

$\frac{z^{2} + 11 z + 30}{z^{2} + z - 20} - \frac{z^{2} + 2 z - 24}{z^{2} + 9 z + 18}$

Step 1

Simplify each term.

Factor $z^{2} + 11 z + 30$ using the AC method.

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 $30$ and whose sum is $11$ .

$5, 6$

Write the factored form using these integers.

$\frac{(z + 5) (z + 6)}{z^{2} + z - 20} - \frac{z^{2} + 2 z - 24}{z^{2} + 9 z + 18}$

Factor $z^{2} + z - 20$ using the AC method.

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$\frac{(z + 5) (z + 6)}{(z - 4) (z + 5)} - \frac{z^{2} + 2 z - 24}{z^{2} + 9 z + 18}$

Cancel the common factor of $z + 5$ .

Cancel the common factor.

$\frac{(z + 5) (z + 6)}{(z - 4) (z + 5)} - \frac{z^{2} + 2 z - 24}{z^{2} + 9 z + 18}$

Rewrite the expression.

$\frac{z + 6}{z - 4} - \frac{z^{2} + 2 z - 24}{z^{2} + 9 z + 18}$

Factor $z^{2} + 2 z - 24$ using the AC method.

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 $- 24$ and whose sum is $2$ .

$- 4, 6$

Write the factored form using these integers.

$\frac{z + 6}{z - 4} - \frac{(z - 4) (z + 6)}{z^{2} + 9 z + 18}$

Factor $z^{2} + 9 z + 18$ using the AC method.

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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$\frac{z + 6}{z - 4} - \frac{(z - 4) (z + 6)}{(z + 3) (z + 6)}$

Cancel the common factor of $z + 6$ .

Cancel the common factor.

$\frac{z + 6}{z - 4} - \frac{(z - 4) (z + 6)}{(z + 3) (z + 6)}$

Rewrite the expression.

$\frac{z + 6}{z - 4} - \frac{z - 4}{z + 3}$

Step 2

To write $\frac{z + 6}{z - 4}$ as a fraction with a common denominator, multiply by $\frac{z + 3}{z + 3}$ .

$\frac{z + 6}{z - 4} \cdot \frac{z + 3}{z + 3} - \frac{z - 4}{z + 3}$

Step 3

To write $- \frac{z - 4}{z + 3}$ as a fraction with a common denominator, multiply by $\frac{z - 4}{z - 4}$ .

$\frac{z + 6}{z - 4} \cdot \frac{z + 3}{z + 3} - \frac{z - 4}{z + 3} \cdot \frac{z - 4}{z - 4}$

Step 4

Write each expression with a common denominator of $(z - 4) (z + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{z + 6}{z - 4}$ by $\frac{z + 3}{z + 3}$ .

$\frac{(z + 6) (z + 3)}{(z - 4) (z + 3)} - \frac{z - 4}{z + 3} \cdot \frac{z - 4}{z - 4}$

Multiply $\frac{z - 4}{z + 3}$ by $\frac{z - 4}{z - 4}$ .

$\frac{(z + 6) (z + 3)}{(z - 4) (z + 3)} - \frac{(z - 4) (z - 4)}{(z + 3) (z - 4)}$

Reorder the factors of $(z - 4) (z + 3)$ .

$\frac{(z + 6) (z + 3)}{(z + 3) (z - 4)} - \frac{(z - 4) (z - 4)}{(z + 3) (z - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(z + 6) (z + 3) - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Step 6

Simplify the numerator.

Expand $(z + 6) (z + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{z (z + 3) + 6 (z + 3) - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Apply the distributive property.

$\frac{z \cdot z + z \cdot 3 + 6 (z + 3) - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Apply the distributive property.

$\frac{z \cdot z + z \cdot 3 + 6 z + 6 \cdot 3 - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ .

$\frac{z^{2} + z \cdot 3 + 6 z + 6 \cdot 3 - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Move $3$ to the left of $z$ .

$\frac{z^{2} + 3 \cdot z + 6 z + 6 \cdot 3 - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Multiply $6$ by $3$ .

$\frac{z^{2} + 3 z + 6 z + 18 - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Add $3 z$ and $6 z$ .

$\frac{z^{2} + 9 z + 18 - (z - 4) (z - 4)}{(z + 3) (z - 4)}$

Apply the distributive property.

$\frac{z^{2} + 9 z + 18 + (- z - - 4) (z - 4)}{(z + 3) (z - 4)}$

Multiply $- 1$ by $- 4$ .

$\frac{z^{2} + 9 z + 18 + (- z + 4) (z - 4)}{(z + 3) (z - 4)}$

Expand $(- z + 4) (z - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{z^{2} + 9 z + 18 - z (z - 4) + 4 (z - 4)}{(z + 3) (z - 4)}$

Apply the distributive property.

$\frac{z^{2} + 9 z + 18 - z \cdot z - z \cdot - 4 + 4 (z - 4)}{(z + 3) (z - 4)}$

Apply the distributive property.

$\frac{z^{2} + 9 z + 18 - z \cdot z - z \cdot - 4 + 4 z + 4 \cdot - 4}{(z + 3) (z - 4)}$

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$\frac{z^{2} + 9 z + 18 - (z \cdot z) - z \cdot - 4 + 4 z + 4 \cdot - 4}{(z + 3) (z - 4)}$

Multiply $z$ by $z$ .

$\frac{z^{2} + 9 z + 18 - z^{2} - z \cdot - 4 + 4 z + 4 \cdot - 4}{(z + 3) (z - 4)}$

Multiply $- 4$ by $- 1$ .

$\frac{z^{2} + 9 z + 18 - z^{2} + 4 z + 4 z + 4 \cdot - 4}{(z + 3) (z - 4)}$

Multiply $4$ by $- 4$ .

$\frac{z^{2} + 9 z + 18 - z^{2} + 4 z + 4 z - 16}{(z + 3) (z - 4)}$

Add $4 z$ and $4 z$ .

$\frac{z^{2} + 9 z + 18 - z^{2} + 8 z - 16}{(z + 3) (z - 4)}$

Subtract $z^{2}$ from $z^{2}$ .

$\frac{9 z + 18 + 0 + 8 z - 16}{(z + 3) (z - 4)}$

Add $9 z$ and $0$ .

$\frac{9 z + 18 + 8 z - 16}{(z + 3) (z - 4)}$

Add $9 z$ and $8 z$ .

$\frac{17 z + 18 - 16}{(z + 3) (z - 4)}$

Subtract $16$ from $18$ .

$\frac{17 z + 2}{(z + 3) (z - 4)}$

Simplify (z^2+11z+30)/(z^2+z-20)-(z^2+2z-24)/(z^2+9z+18)

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