Basic Math Examples

$\frac{z^{2} - z - 6}{3 z - 9} \cdot \frac{z^{2} - 9}{z^{2} + 6 z + 9}$

Step 1

Factor $z^{2} - z - 6$ using the AC method.

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Step 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(z - 3) (z + 2)}{3 z - 9} \cdot \frac{z^{2} - 9}{z^{2} + 6 z + 9}$

Step 2

Factor $3$ out of $3 z - 9$ .

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Step 2.1

Factor $3$ out of $3 z$ .

$\frac{(z - 3) (z + 2)}{3 (z) - 9} \cdot \frac{z^{2} - 9}{z^{2} + 6 z + 9}$

Step 2.2

Factor $3$ out of $- 9$ .

$\frac{(z - 3) (z + 2)}{3 z + 3 \cdot - 3} \cdot \frac{z^{2} - 9}{z^{2} + 6 z + 9}$

Step 2.3

Factor $3$ out of $3 z + 3 \cdot - 3$ .

$\frac{(z - 3) (z + 2)}{3 (z - 3)} \cdot \frac{z^{2} - 9}{z^{2} + 6 z + 9}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{(z - 3) (z + 2)}{3 (z - 3)} \cdot \frac{z^{2} - 3^{2}}{z^{2} + 6 z + 9}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 3$ .

$\frac{(z - 3) (z + 2)}{3 (z - 3)} \cdot \frac{(z + 3) (z - 3)}{z^{2} + 6 z + 9}$

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.2

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

$6 z = 2 \cdot z \cdot 3$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 3$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine.

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

Step 5.2

Cancel the common factor of $z - 3$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Cancel the common factor of $z + 3$ and ${(z + 3)}^{2}$ .

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Step 5.3.1

Factor $z + 3$ out of $(z + 2) ((z + 3) (z - 3))$ .

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

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $z + 3$ out of $3 {(z + 3)}^{2}$ .

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

Step 5.3.2.2

Cancel the common factor.

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

Step 5.3.2.3

Rewrite the expression.

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