Basic Math Examples

$12 y + 9 y^{2} - 21$

Step 1

Factor $3$ out of $12 y + 9 y^{2} - 21$ .

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Step 1.1

Factor $3$ out of $12 y$ .

$3 (4 y) + 9 y^{2} - 21$

Step 1.2

Factor $3$ out of $9 y^{2}$ .

$3 (4 y) + 3 (3 y^{2}) - 21$

Step 1.3

Factor $3$ out of $- 21$ .

$3 (4 y) + 3 (3 y^{2}) + 3 (- 7)$

Step 1.4

Factor $3$ out of $3 (4 y) + 3 (3 y^{2})$ .

$3 (4 y + 3 y^{2}) + 3 (- 7)$

Step 1.5

Factor $3$ out of $3 (4 y + 3 y^{2}) + 3 (- 7)$ .

$3 (4 y + 3 y^{2} - 7)$

Step 2

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$3 (4 u + 3 u^{2} - 7)$

Step 3

Factor by grouping.

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Step 3.1

Reorder terms.

$3 (3 u^{2} + 4 u - 7)$

Step 3.2

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 $a \cdot c = 3 \cdot - 7 = - 21$ and whose sum is $b = 4$ .

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Step 3.2.1

Factor $4$ out of $4 u$ .

$3 (3 u^{2} + 4 (u) - 7)$

Step 3.2.2

Rewrite $4$ as $- 3$ plus $7$

$3 (3 u^{2} + (- 3 + 7) u - 7)$

Step 3.2.3

Apply the distributive property.

$3 (3 u^{2} - 3 u + 7 u - 7)$

Step 3.3

Factor out the greatest common factor from each group.

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Step 3.3.1

Group the first two terms and the last two terms.

$3 ((3 u^{2} - 3 u) + 7 u - 7)$

Step 3.3.2

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

$3 (3 u (u - 1) + 7 (u - 1))$

Step 3.4

Factor the polynomial by factoring out the greatest common factor, $u - 1$ .

$3 ((u - 1) (3 u + 7))$

Step 4

Factor.

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Step 4.1

Replace all occurrences of $u$ with $y$ .

$3 ((y - 1) (3 y + 7))$

Step 4.2

Remove unnecessary parentheses.

$3 (y - 1) (3 y + 7)$