Basic Math Examples

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

Step 1

Cancel the common factor of $2 + z$ and $z + 2$ .

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Step 1.1

Reorder terms.

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

Step 1.2

Cancel the common factor.

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

Step 1.3

Rewrite the expression.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

$(z^{2} + 3 z + 2) (z + 1)$

Step 3

Expand $(z^{2} + 3 z + 2) (z + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 4.1.1.1

Multiply $z^{2}$ by $z$ .

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Step 4.1.1.1.1

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

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

Step 4.1.1.1.2

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

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

Step 4.1.1.2

Add $2$ and $1$ .

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

Step 4.1.2

Multiply $z^{2}$ by $1$ .

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

Step 4.1.3

Multiply $z$ by $z$ by adding the exponents.

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Step 4.1.3.1

Move $z$ .

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

Step 4.1.3.2

Multiply $z$ by $z$ .

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

Step 4.1.4

Multiply $3$ by $1$ .

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

Step 4.1.5

Multiply $2$ by $1$ .

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

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Add $z^{2}$ and $3 z^{2}$ .

$z^{3} + 4 z^{2} + 3 z + 2 z + 2$

Step 4.2.2

Add $3 z$ and $2 z$ .

$z^{3} + 4 z^{2} + 5 z + 2$