Basic Math Examples

$\frac{z^{2} + 7 z}{3 z + 21} \cdot \frac{z^{2} + 3 z + 2}{z + 2}$

Step 1

Simplify with factoring out.

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Step 1.1

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

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Step 1.1.1

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

$\frac{z \cdot z + 7 z}{3 z + 21} \cdot \frac{z^{2} + 3 z + 2}{z + 2}$

Step 1.1.2

Factor $z$ out of $7 z$ .

$\frac{z \cdot z + z \cdot 7}{3 z + 21} \cdot \frac{z^{2} + 3 z + 2}{z + 2}$

Step 1.1.3

Factor $z$ out of $z \cdot z + z \cdot 7$ .

$\frac{z (z + 7)}{3 z + 21} \cdot \frac{z^{2} + 3 z + 2}{z + 2}$

Step 1.2

Factor $3$ out of $3 z + 21$ .

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Step 1.2.1

Factor $3$ out of $3 z$ .

$\frac{z (z + 7)}{3 (z) + 21} \cdot \frac{z^{2} + 3 z + 2}{z + 2}$

Step 1.2.2

Factor $3$ out of $21$ .

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

Step 1.2.3

Factor $3$ out of $3 z + 3 \cdot 7$ .

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

Step 2

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

$1, 2$

Step 2.2

Write the factored form using these integers.

$\frac{z (z + 7)}{3 (z + 7)} \cdot \frac{(z + 1) (z + 2)}{z + 2}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $z + 7$ .

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Step 3.1.1

Cancel the common factor.

$\frac{z (z + 7)}{3 (z + 7)} \cdot \frac{(z + 1) (z + 2)}{z + 2}$

Step 3.1.2

Rewrite the expression.

$\frac{z}{3} \cdot \frac{(z + 1) (z + 2)}{z + 2}$

Step 3.2

Cancel the common factor of $z + 2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{z}{3} \cdot \frac{(z + 1) (z + 2)}{z + 2}$

Step 3.2.2

Divide $z + 1$ by $1$ .

$\frac{z}{3} \cdot (z + 1)$

Step 3.3

Apply the distributive property.

$\frac{z}{3} z + \frac{z}{3} \cdot 1$

Step 4

Multiply $\frac{z}{3} z$ .

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Step 4.1

Combine $\frac{z}{3}$ and $z$ .

$\frac{z \cdot z}{3} + \frac{z}{3} \cdot 1$

Step 4.2

Raise $z$ to the power of $1$ .

$\frac{z^{1} z}{3} + \frac{z}{3} \cdot 1$

Step 4.3

Raise $z$ to the power of $1$ .

$\frac{z^{1} z^{1}}{3} + \frac{z}{3} \cdot 1$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{z^{1 + 1}}{3} + \frac{z}{3} \cdot 1$

Step 4.5

Add $1$ and $1$ .

$\frac{z^{2}}{3} + \frac{z}{3} \cdot 1$

Step 5

Multiply $\frac{z}{3}$ by $1$ .

$\frac{z^{2}}{3} + \frac{z}{3}$

Step 6

Combine the numerators over the common denominator.

$\frac{z^{2} + z}{3}$

Step 7

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

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Step 7.1

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

$\frac{z \cdot z + z}{3}$

Step 7.2

Raise $z$ to the power of $1$ .

$\frac{z \cdot z + z^{1}}{3}$

Step 7.3

Factor $z$ out of $z^{1}$ .

$\frac{z \cdot z + z \cdot 1}{3}$

Step 7.4

Factor $z$ out of $z \cdot z + z \cdot 1$ .

$\frac{z (z + 1)}{3}$