Basic Math Examples

$\frac{z^{2} + 14 z + 49}{z - 3} \div \frac{z^{2} + 13 z + 42}{z^{2} + 11 z + 30}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 14 z + 49}{z - 3} \cdot \frac{z^{2} + 11 z + 30}{z^{2} + 13 z + 42}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{z^{2} + 14 z + 7^{2}}{z - 3} \cdot \frac{z^{2} + 11 z + 30}{z^{2} + 13 z + 42}$

Step 2.2

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

$14 z = 2 \cdot z \cdot 7$

Step 2.3

Rewrite the polynomial.

$\frac{z^{2} + 2 \cdot z \cdot 7 + 7^{2}}{z - 3} \cdot \frac{z^{2} + 11 z + 30}{z^{2} + 13 z + 42}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 7$ .

$\frac{{(z + 7)}^{2}}{z - 3} \cdot \frac{z^{2} + 11 z + 30}{z^{2} + 13 z + 42}$

Step 3

Factor $z^{2} + 11 z + 30$ 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 $30$ and whose sum is $11$ .

$5, 6$

Step 3.2

Write the factored form using these integers.

$\frac{{(z + 7)}^{2}}{z - 3} \cdot \frac{(z + 5) (z + 6)}{z^{2} + 13 z + 42}$

Step 4

Factor $z^{2} + 13 z + 42$ using the AC method.

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Step 4.1

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 $42$ and whose sum is $13$ .

$6, 7$

Step 4.2

Write the factored form using these integers.

$\frac{{(z + 7)}^{2}}{z - 3} \cdot \frac{(z + 5) (z + 6)}{(z + 6) (z + 7)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $z + 7$ .

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Step 5.1.1

Factor $z + 7$ out of ${(z + 7)}^{2}$ .

$\frac{(z + 7) (z + 7)}{z - 3} \cdot \frac{(z + 5) (z + 6)}{(z + 6) (z + 7)}$

Step 5.1.2

Factor $z + 7$ out of $(z + 6) (z + 7)$ .

$\frac{(z + 7) (z + 7)}{z - 3} \cdot \frac{(z + 5) (z + 6)}{(z + 7) (z + 6)}$

Step 5.1.3

Cancel the common factor.

$\frac{(z + 7) (z + 7)}{z - 3} \cdot \frac{(z + 5) (z + 6)}{(z + 7) (z + 6)}$

Step 5.1.4

Rewrite the expression.

$\frac{z + 7}{z - 3} \cdot \frac{(z + 5) (z + 6)}{z + 6}$

Step 5.2

Multiply $\frac{z + 7}{z - 3}$ by $\frac{(z + 5) (z + 6)}{z + 6}$ .

$\frac{(z + 7) ((z + 5) (z + 6))}{(z - 3) (z + 6)}$

Step 5.3

Cancel the common factor of $z + 6$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(z + 7) ((z + 5) (z + 6))}{(z - 3) (z + 6)}$

Step 5.3.2

Rewrite the expression.

$\frac{(z + 7) (z + 5)}{z - 3}$