Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 1.1.1.1

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

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

Step 1.1.1.2

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.1.4

Multiply $x$ by $1$ .

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

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Raise $x - 1$ to the power of $1$ .

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

Step 4.3

Raise $x - 1$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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

Step 4.7

Reorder the factors of $(2 x + 1) {(x - 1)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $- 1$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Rewrite using the commutative property of multiplication.

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

Step 6.5

Multiply $3$ by $1$ .

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

Step 6.6

Simplify each term.

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Step 6.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.6.1.1

Move $x$ .

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

Step 6.6.1.2

Multiply $x$ by $x$ .

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

Step 6.6.2

Multiply $3$ by $2$ .

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

Step 6.7

Add $2 x$ and $3 x$ .

$\frac{5 x - 2 + 6 x^{2}}{{(x - 1)}^{2} (2 x + 1)}$

Step 6.8

Reorder terms.

$\frac{6 x^{2} + 5 x - 2}{{(x - 1)}^{2} (2 x + 1)}$