Precalculus Examples

$\frac{2 x^{2} + 5 x - 3}{1 - 2 x}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

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Step 1.1.1

Factor $5$ out of $5 x$ .

$\frac{2 x^{2} + 5 (x) - 3}{1 - 2 x}$

Step 1.1.2

Rewrite $5$ as $- 1$ plus $6$

$\frac{2 x^{2} + (- 1 + 6) x - 3}{1 - 2 x}$

Step 1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x + 6 x - 3}{1 - 2 x}$

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.

$\frac{(2 x^{2} - 1 x) + 6 x - 3}{1 - 2 x}$

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

Cancel the common factor of $2 x - 1$ and $1 - 2 x$ .

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Step 2.1

Factor $- 1$ out of $2 x$ .

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

Step 2.2

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

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

Step 2.3

Factor $- 1$ out of $- (- 2 x) - 1 (1)$ .

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

Step 2.4

Rewrite $- (- 2 x + 1)$ as $- 1 (- 2 x + 1)$ .

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

Step 2.5

Reorder terms.

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

Step 2.6

Cancel the common factor.

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

Step 2.7

Divide $- 1 (x + 3)$ by $1$ .

$- 1 (x + 3)$

Step 3

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

$- (x + 3)$

Step 4

Apply the distributive property.

$- x - 1 \cdot 3$

Step 5

Multiply $- 1$ by $3$ .

$- x - 3$