Basic Math Examples

$\frac{\frac{5 y^{2} - 34 y - 7}{y + 2}}{\frac{5 y^{2} + 11 y + 2}{y^{2} + 4 y + 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 y^{2} - 34 y - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 7 = - 35$ and whose sum is $b = - 34$ .

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Step 2.1.1

Factor $- 34$ out of $- 34 y$ .

$\frac{5 y^{2} - 34 (y) - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.1.2

Rewrite $- 34$ as $1$ plus $- 35$

$\frac{5 y^{2} + (1 - 35) y - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.1.3

Apply the distributive property.

$\frac{5 y^{2} + 1 y - 35 y - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.1.4

Multiply $y$ by $1$ .

$\frac{5 y^{2} + y - 35 y - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(5 y^{2} + y) - 35 y - 7}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{y (5 y + 1) - 7 (5 y + 1)}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 y + 1$ .

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{y^{2} + 4 y + 4}{5 y^{2} + 11 y + 2}$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{y^{2} + 4 y + 2^{2}}{5 y^{2} + 11 y + 2}$

Step 3.2

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

$4 y = 2 \cdot y \cdot 2$

Step 3.3

Rewrite the polynomial.

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{y^{2} + 2 \cdot y \cdot 2 + 2^{2}}{5 y^{2} + 11 y + 2}$

Step 3.4

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

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{5 y^{2} + 11 y + 2}$

Step 4

Factor by grouping.

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Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 2 = 10$ and whose sum is $b = 11$ .

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Step 4.1.1

Factor $11$ out of $11 y$ .

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{5 y^{2} + 11 (y) + 2}$

Step 4.1.2

Rewrite $11$ as $1$ plus $10$

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{5 y^{2} + (1 + 10) y + 2}$

Step 4.1.3

Apply the distributive property.

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{5 y^{2} + 1 y + 10 y + 2}$

Step 4.1.4

Multiply $y$ by $1$ .

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{5 y^{2} + y + 10 y + 2}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{(5 y^{2} + y) + 10 y + 2}$

Step 4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{y (5 y + 1) + 2 (5 y + 1)}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $5 y + 1$ .

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{(5 y + 1) (y + 2)}$

Step 5

Cancel the common factor of $5 y + 1$ .

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Step 5.1

Cancel the common factor.

$\frac{(5 y + 1) (y - 7)}{y + 2} \cdot \frac{{(y + 2)}^{2}}{(5 y + 1) (y + 2)}$

Step 5.2

Rewrite the expression.

$\frac{y - 7}{y + 2} \cdot \frac{{(y + 2)}^{2}}{y + 2}$

Step 6

Cancel the common factor of $y + 2$ .

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Step 6.1

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

$\frac{y - 7}{y + 2} \cdot \frac{(y + 2) (y + 2)}{y + 2}$

Step 6.2

Cancel the common factor.

$\frac{y - 7}{y + 2} \cdot \frac{(y + 2) (y + 2)}{y + 2}$

Step 6.3

Rewrite the expression.

$(y - 7) \frac{y + 2}{y + 2}$

Step 7

Cancel the common factor of $y + 2$ .

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Step 7.1

Cancel the common factor.

$(y - 7) \frac{y + 2}{y + 2}$

Step 7.2

Rewrite the expression.

$(y - 7) \cdot 1$

Step 8

Multiply $y - 7$ by $1$ .

$y - 7$