Basic Math Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.2

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

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Step 1.2.1

Factor $3$ out of $3 z$ .

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

Step 1.2.2

Factor $3$ out of $3$ .

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

Step 1.2.3

Factor $3$ out of $3 (z) + 3 (1)$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.2

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

$2 y = 2 \cdot y \cdot 1$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 1$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $y + 1$ out of $3 z {(y + 1)}^{2}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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