Algebra Examples

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

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 $- 5 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Apply the distributive property.

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

Step 1.1.4

Multiply $x$ by $1$ .

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

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.

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

To write $- \frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

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

Step 3

Write each expression with a common denominator of $(2 x + 1) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x - 3}$ by $\frac{2 x + 1}{2 x + 1}$ .

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

Step 3.2

Reorder the factors of $(x - 3) (2 x + 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Subtract $2 x$ from $3 x$ .

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

Step 5.5

Subtract $1$ from $- 2$ .

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

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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