Pre-Algebra Examples

$\frac{z^{2} + 11 z + 18}{z^{2} + 14 z + 45} \div \frac{z^{2} + 2 z}{z^{2} + 10 z + 25}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 11 z + 18}{z^{2} + 14 z + 45} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Step 2

Factor $z^{2} + 11 z + 18$ using the AC method.

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Step 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 $18$ and whose sum is $11$ .

$2, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{z^{2} + 14 z + 45} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Step 3

Factor $z^{2} + 14 z + 45$ using the AC method.

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Step 3.1

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 $45$ and whose sum is $14$ .

$5, 9$

Step 3.2

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Step 4

Cancel the common factor of $z + 9$ .

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Step 4.1

Cancel the common factor.

$\frac{(z + 2) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Step 4.2

Rewrite the expression.

$\frac{z + 2}{z + 5} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $25$ as $5^{2}$ .

$\frac{z + 2}{z + 5} \cdot \frac{z^{2} + 10 z + 5^{2}}{z^{2} + 2 z}$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 z = 2 \cdot z \cdot 5$

Step 5.3

Rewrite the polynomial.

$\frac{z + 2}{z + 5} \cdot \frac{z^{2} + 2 \cdot z \cdot 5 + 5^{2}}{z^{2} + 2 z}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 5$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z^{2} + 2 z}$

Step 6

Factor $z$ out of $z^{2} + 2 z$ .

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Step 6.1

Factor $z$ out of $z^{2}$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z \cdot z + 2 z}$

Step 6.2

Factor $z$ out of $2 z$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z \cdot z + z \cdot 2}$

Step 6.3

Factor $z$ out of $z \cdot z + z \cdot 2$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z (z + 2)}$

Step 7

Cancel the common factor of $z + 2$ .

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Step 7.1

Factor $z + 2$ out of $z (z + 2)$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{(z + 2) z}$

Step 7.2

Cancel the common factor.

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{(z + 2) z}$

Step 7.3

Rewrite the expression.

$\frac{1}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z}$

Step 8

Cancel the common factor of $z + 5$ .

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Step 8.1

Factor $z + 5$ out of ${(z + 5)}^{2}$ .

$\frac{1}{z + 5} \cdot \frac{(z + 5) (z + 5)}{z}$

Step 8.2

Cancel the common factor.

$\frac{1}{z + 5} \cdot \frac{(z + 5) (z + 5)}{z}$

Step 8.3

Rewrite the expression.

$\frac{z + 5}{z}$

Step 9

Split the fraction $\frac{z + 5}{z}$ into two fractions.

$\frac{z}{z} + \frac{5}{z}$

Step 10

Cancel the common factor of $z$ .

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Step 10.1

Cancel the common factor.

$\frac{z}{z} + \frac{5}{z}$

Step 10.2

Rewrite the expression.

$1 + \frac{5}{z}$