Algebra Examples

$9 - 6 a - 24 a^{2}$

Step 1

Factor out the GCF of $3$ from $9 - 6 a - 24 a^{2}$ .

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Step 1.1

Factor out the GCF of $3$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of $3$ from the expression $9$ .

$3 (3) - 6 a - 24 a^{2}$

Step 1.1.2

Factor out the GCF of $3$ from the expression $- 6 a$ .

$3 (3) + 3 (- 2 a) - 24 a^{2}$

Step 1.1.3

Factor out the GCF of $3$ from the expression $- 24 a^{2}$ .

$3 (3) + 3 (- 2 a) + 3 (- 8 a^{2})$

Step 1.2

Since all the terms share a common factor of $3$ , it can be factored out of each term.

$3 (3 - 2 a - 8 a^{2})$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$3 (- 8 a^{2} - 2 a + 3)$

Step 2.2

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 $a \cdot c = - 8 \cdot 3 = - 24$ and whose sum is $b = - 2$ .

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Step 2.2.1

Factor $- 2$ out of $- 2 a$ .

$3 (- 8 a^{2} - 2 a + 3)$

Step 2.2.2

Rewrite $- 2$ as $4$ plus $- 6$

$3 (- 8 a^{2} + (4 - 6) a + 3)$

Step 2.2.3

Apply the distributive property.

$3 (- 8 a^{2} + 4 a - 6 a + 3)$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$3 ((- 8 a^{2} + 4 a) - 6 a + 3)$

Step 2.3.2

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

$3 (4 a (- 2 a + 1) + 3 (- 2 a + 1))$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $- 2 a + 1$ .

$3 ((- 2 a + 1) (4 a + 3))$