Precalculus Examples

$\frac{2 x^{2} - 9 x + 4}{12 + x - x^{2}}$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 2 \cdot 4 = 8$ and whose sum is $b = - 9$ .

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Step 1.1.1

Factor $- 9$ out of $- 9 x$ .

$\frac{2 x^{2} - 9 (x) + 4}{12 + x - x^{2}}$

Step 1.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$\frac{2 x^{2} + (- 1 - 8) x + 4}{12 + x - x^{2}}$

Step 1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x - 8 x + 4}{12 + x - x^{2}}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 1 x) - 8 x + 4}{12 + x - x^{2}}$

Step 1.2.2

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

$\frac{x (2 x - 1) - 4 (2 x - 1)}{12 + x - x^{2}}$

Step 1.3

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

$\frac{(2 x - 1) (x - 4)}{12 + x - x^{2}}$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$\frac{(2 x - 1) (x - 4)}{- x^{2} + x + 12}$

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = 1$ .

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Step 2.2.1

Multiply by $1$ .

$\frac{(2 x - 1) (x - 4)}{- x^{2} + 1 x + 12}$

Step 2.2.2

Rewrite $1$ as $- 3$ plus $4$

$\frac{(2 x - 1) (x - 4)}{- x^{2} + (- 3 + 4) x + 12}$

Step 2.2.3

Apply the distributive property.

$\frac{(2 x - 1) (x - 4)}{- x^{2} - 3 x + 4 x + 12}$

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.

$\frac{(2 x - 1) (x - 4)}{(- x^{2} - 3 x) + 4 x + 12}$

Step 2.3.2

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

$\frac{(2 x - 1) (x - 4)}{x (- x - 3) - 4 (- x - 3)}$

Step 2.4

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

$\frac{(2 x - 1) (x - 4)}{(- x - 3) (x - 4)}$

Step 3

Cancel the common factor of $x - 4$ .

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Step 3.1

Cancel the common factor.

$\frac{(2 x - 1) (x - 4)}{(- x - 3) (x - 4)}$

Step 3.2

Rewrite the expression.

$\frac{2 x - 1}{- x - 3}$

Step 4

Factor $- 1$ out of $- x$ .

$\frac{2 x - 1}{- (x) - 3}$

Step 5

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{2 x - 1}{- (x) - 1 (3)}$

Step 6

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{2 x - 1}{- (x + 3)}$

Step 7

Rewrite negatives.

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Step 7.1

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{2 x - 1}{- 1 (x + 3)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{2 x - 1}{x + 3}$