Pre-Algebra Examples

$\frac{z^{2} - 4}{z^{2}} \cdot \frac{z^{2} - 2 z}{z^{2} + 3 z - 10}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{z^{2} - 2^{2}}{z^{2}} \cdot \frac{z^{2} - 2 z}{z^{2} + 3 z - 10}$

Step 1.2

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 $b = 2$ .

$\frac{(z + 2) (z - 2)}{z^{2}} \cdot \frac{z^{2} - 2 z}{z^{2} + 3 z - 10}$

Step 2

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

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Step 2.1

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

$\frac{(z + 2) (z - 2)}{z^{2}} \cdot \frac{z \cdot z - 2 z}{z^{2} + 3 z - 10}$

Step 2.2

Factor $z$ out of $- 2 z$ .

$\frac{(z + 2) (z - 2)}{z^{2}} \cdot \frac{z \cdot z + z \cdot - 2}{z^{2} + 3 z - 10}$

Step 2.3

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

$\frac{(z + 2) (z - 2)}{z^{2}} \cdot \frac{z (z - 2)}{z^{2} + 3 z - 10}$

Step 3

Factor $z^{2} + 3 z - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $z - 2$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $z$ and $z^{2}$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

$\frac{z ((z + 2) (z - 2))}{z (z (z + 5))}$

Step 4.3.2.2

Cancel the common factor.

$\frac{z ((z + 2) (z - 2))}{z (z (z + 5))}$

Step 4.3.2.3

Rewrite the expression.

$\frac{(z + 2) (z - 2)}{z (z + 5)}$