Basic Math Examples

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{y^{2} + 6 y + 9}{5 y^{2} + 17 y + 6}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{y^{2} + 6 y + 3^{2}}{5 y^{2} + 17 y + 6}$

Step 1.2

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

$6 y = 2 \cdot y \cdot 3$

Step 1.3

Rewrite the polynomial.

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{y^{2} + 2 \cdot y \cdot 3 + 3^{2}}{5 y^{2} + 17 y + 6}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 3$ .

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{5 y^{2} + 17 y + 6}$

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 $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = 17$ .

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Step 2.1.1

Factor $17$ out of $17 y$ .

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{5 y^{2} + 17 (y) + 6}$

Step 2.1.2

Rewrite $17$ as $2$ plus $15$

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{5 y^{2} + (2 + 15) y + 6}$

Step 2.1.3

Apply the distributive property.

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{5 y^{2} + 2 y + 15 y + 6}$

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.

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{(5 y^{2} + 2 y) + 15 y + 6}$

Step 2.2.2

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

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{y (5 y + 2) + 3 (5 y + 2)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 y + 2$ .

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{{(y + 3)}^{2}}{(5 y + 2) (y + 3)}$

Step 3

Cancel the common factor of $y + 3$ .

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Step 3.1

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

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{(y + 3) (y + 3)}{(5 y + 2) (y + 3)}$

Step 3.2

Cancel the common factor.

$\frac{5 y^{2} - 33 y + 14}{y + 3} \cdot \frac{(y + 3) (y + 3)}{(5 y + 2) (y + 3)}$

Step 3.3

Rewrite the expression.

$(5 y^{2} - 33 y + 14) \cdot \frac{y + 3}{(5 y + 2) (y + 3)}$

Step 4

Cancel the common factor of $y + 3$ .

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Step 4.1

Cancel the common factor.

$(5 y^{2} - 33 y + 14) \cdot \frac{y + 3}{(5 y + 2) (y + 3)}$

Step 4.2

Rewrite the expression.

$(5 y^{2} - 33 y + 14) \cdot \frac{1}{5 y + 2}$

Step 5

Multiply $5 y^{2} - 33 y + 14$ by $\frac{1}{5 y + 2}$ .

$\frac{5 y^{2} - 33 y + 14}{5 y + 2}$