Simplify (x^2+2x+1)/(x^3-2x^2-3x)

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

Step 1

Factor using the perfect square rule.

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

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

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

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

Rewrite the polynomial.

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

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

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

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - 2 x^{2} - 3 x$ .

Factor $x$ out of $x^{3}$ .

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

Factor $x$ out of $- 2 x^{2}$ .

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

Factor $x$ out of $- 3 x$ .

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

Factor $x$ out of $x \cdot x^{2} + x (- 2 x)$ .

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

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot - 3$ .

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

Factor $x^{2} - 2 x - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

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

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x^2+2x+1)/(x^3-2x^2-3x)

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