Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.3

Multiply $- 1$ by $3$ .

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

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Combine the opposite terms in $x^{2} + 3 x + 2 - 3 x - 3$ .

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Step 1.3.1.1

Subtract $3 x$ from $3 x$ .

$\frac{x^{2} + 0 + 2 - 3}{x^{2} - 2 x + 1}$

Step 1.3.1.2

Add $x^{2}$ and $0$ .

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

Step 1.3.2

Subtract $3$ from $2$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

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 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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