Pre-Algebra Examples

$\frac{z^{2} - 3 z - 18}{7 z^{3} - 4 z^{2}} \cdot \frac{x (49 z^{3} - 16 z)}{6 z - 36}$

Step 1

Factor $z^{2} - 3 z - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(z - 6) (z + 3)}{7 z^{3} - 4 z^{2}} \cdot \frac{x (49 z^{3} - 16 z)}{6 z - 36}$

Step 2

Factor $z^{2}$ out of $7 z^{3} - 4 z^{2}$ .

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Step 2.1

Factor $z^{2}$ out of $7 z^{3}$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z) - 4 z^{2}} \cdot \frac{x (49 z^{3} - 16 z)}{6 z - 36}$

Step 2.2

Factor $z^{2}$ out of $- 4 z^{2}$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z) + z^{2} \cdot - 4} \cdot \frac{x (49 z^{3} - 16 z)}{6 z - 36}$

Step 2.3

Factor $z^{2}$ out of $z^{2} (7 z) + z^{2} \cdot - 4$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (49 z^{3} - 16 z)}{6 z - 36}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $z$ out of $49 z^{3} - 16 z$ .

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Step 3.1.1

Factor $z$ out of $49 z^{3}$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (z (49 z^{2}) - 16 z)}{6 z - 36}$

Step 3.1.2

Factor $z$ out of $- 16 z$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (z (49 z^{2}) + z \cdot - 16)}{6 z - 36}$

Step 3.1.3

Factor $z$ out of $z (49 z^{2}) + z \cdot - 16$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (z (49 z^{2} - 16))}{6 z - 36}$

Step 3.2

Rewrite $49 z^{2}$ as ${(7 z)}^{2}$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (z ({(7 z)}^{2} - 16))}{6 z - 36}$

Step 3.3

Rewrite $16$ as $4^{2}$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x (z ({(7 z)}^{2} - 4^{2}))}{6 z - 36}$

Step 3.4

Factor.

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x z (7 z + 4) (7 z - 4)}{6 z - 36}$

Step 4

Simplify terms.

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Step 4.1

Factor $6$ out of $6 z - 36$ .

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Step 4.1.1

Factor $6$ out of $6 z$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x z (7 z + 4) (7 z - 4)}{6 (z) - 36}$

Step 4.1.2

Factor $6$ out of $- 36$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x z (7 z + 4) (7 z - 4)}{6 z + 6 \cdot - 6}$

Step 4.1.3

Factor $6$ out of $6 z + 6 \cdot - 6$ .

$\frac{(z - 6) (z + 3)}{z^{2} (7 z - 4)} \cdot \frac{x z (7 z + 4) (7 z - 4)}{6 (z - 6)}$

Step 4.2

Combine.

$\frac{(z - 6) (z + 3) (x z (7 z + 4) (7 z - 4))}{z^{2} (7 z - 4) (6 (z - 6))}$

Step 4.3

Cancel the common factor of $z - 6$ .

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Step 4.3.1

Cancel the common factor.

$\frac{(z - 6) (z + 3) (x z (7 z + 4) (7 z - 4))}{z^{2} (7 z - 4) (6 (z - 6))}$

Step 4.3.2

Rewrite the expression.

$\frac{(z + 3) (x z (7 z + 4) (7 z - 4))}{z^{2} (7 z - 4) \cdot (6)}$

Step 4.4

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

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Step 4.4.1

Factor $z$ out of $(z + 3) (x z (7 z + 4) (7 z - 4))$ .

$\frac{z ((z + 3) ((x (7 z + 4)) (7 z - 4)))}{z^{2} (7 z - 4) \cdot (6)}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $z$ out of $z^{2} (7 z - 4) \cdot (6)$ .

$\frac{z ((z + 3) ((x (7 z + 4)) (7 z - 4)))}{z ((z (7 z - 4)) \cdot 6)}$

Step 4.4.2.2

Cancel the common factor.

$\frac{z ((z + 3) ((x (7 z + 4)) (7 z - 4)))}{z ((z (7 z - 4)) \cdot 6)}$

Step 4.4.2.3

Rewrite the expression.

$\frac{(z + 3) ((x (7 z + 4)) (7 z - 4))}{(z (7 z - 4)) \cdot 6}$

Step 4.5

Cancel the common factor of $7 z - 4$ .

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Step 4.5.1

Cancel the common factor.

$\frac{(z + 3) (x (7 z + 4) (7 z - 4))}{z (7 z - 4) \cdot 6}$

Step 4.5.2

Rewrite the expression.

$\frac{(z + 3) (x (7 z + 4))}{(z) \cdot 6}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Move $6$ to the left of $z$ .

$\frac{(z + 3) x (7 z + 4)}{6 \cdot z}$

Step 4.6.2

Reorder factors in $\frac{(z + 3) x (7 z + 4)}{6 z}$ .

$\frac{x (z + 3) (7 z + 4)}{6 z}$