Precalculus Examples

$12 x^{3} - 34 x^{2} + 10 x$

Step 1

Factor $2 x$ out of $12 x^{3} - 34 x^{2} + 10 x$ .

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Step 1.1

Factor $2 x$ out of $12 x^{3}$ .

$2 x (6 x^{2}) - 34 x^{2} + 10 x$

Step 1.2

Factor $2 x$ out of $- 34 x^{2}$ .

$2 x (6 x^{2}) + 2 x (- 17 x) + 10 x$

Step 1.3

Factor $2 x$ out of $10 x$ .

$2 x (6 x^{2}) + 2 x (- 17 x) + 2 x (5)$

Step 1.4

Factor $2 x$ out of $2 x (6 x^{2}) + 2 x (- 17 x)$ .

$2 x (6 x^{2} - 17 x) + 2 x (5)$

Step 1.5

Factor $2 x$ out of $2 x (6 x^{2} - 17 x) + 2 x (5)$ .

$2 x (6 x^{2} - 17 x + 5)$

Step 2

Factor.

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Step 2.1

Factor by grouping.

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Step 2.1.1

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

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Step 2.1.1.1

Factor $- 17$ out of $- 17 x$ .

$2 x (6 x^{2} - 17 (x) + 5)$

Step 2.1.1.2

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

$2 x (6 x^{2} + (- 2 - 15) x + 5)$

Step 2.1.1.3

Apply the distributive property.

$2 x (6 x^{2} - 2 x - 15 x + 5)$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$2 x ((6 x^{2} - 2 x) - 15 x + 5)$

Step 2.1.2.2

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

$2 x (2 x (3 x - 1) - 5 (3 x - 1))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$2 x ((3 x - 1) (2 x - 5))$

Step 2.2

Remove unnecessary parentheses.

$2 x (3 x - 1) (2 x - 5)$