Pre-Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 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 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

Split the fraction $\frac{x^{3} - x^{2} - 5 x - 3}{{(x + 1)}^{2}}$ into two fractions.

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

Step 4

Split the fraction $\frac{x^{3} - x^{2} - 5 x}{{(x + 1)}^{2}}$ into two fractions.

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

Step 5

Split the fraction $\frac{x^{3} - x^{2}}{{(x + 1)}^{2}}$ into two fractions.

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

Step 6

Move the negative in front of the fraction.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Move the negative in front of the fraction.

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