Basic Math Examples

$\frac{a^{3} - 2 a^{2} + a}{a^{2} - a}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a$ out of $a^{3} - 2 a^{2} + a$ .

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Step 1.1.1

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

$\frac{a \cdot a^{2} - 2 a^{2} + a}{a^{2} - a}$

Step 1.1.2

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

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

Step 1.1.3

Raise $a$ to the power of $1$ .

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

Step 1.1.4

Factor $a$ out of $a^{1}$ .

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

Step 1.1.5

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

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

Step 1.1.6

Factor $a$ out of $a (a^{2} - 2 a) + a \cdot 1$ .

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

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

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

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.

$2 a = 2 \cdot a \cdot 1$

Step 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

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

$\frac{a {(a - 1)}^{2}}{a^{2} - a}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $a$ out of $a^{2}$ .

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

Step 2.1.2

Factor $a$ out of $- a$ .

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

Step 2.1.3

Factor $a$ out of $a \cdot a + a \cdot - 1$ .

$\frac{a {(a - 1)}^{2}}{a (a - 1)}$

Step 2.2

Cancel the common factor of $a$ .

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Step 2.2.1

Cancel the common factor.

$\frac{a {(a - 1)}^{2}}{a (a - 1)}$

Step 2.2.2

Rewrite the expression.

$\frac{{(a - 1)}^{2}}{a - 1}$

Step 2.3

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

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Step 2.3.1

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

$\frac{(a - 1) (a - 1)}{a - 1}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Multiply by $1$ .

$\frac{(a - 1) (a - 1)}{(a - 1) \cdot 1}$

Step 2.3.2.2

Cancel the common factor.

$\frac{(a - 1) (a - 1)}{(a - 1) \cdot 1}$

Step 2.3.2.3

Rewrite the expression.

$\frac{a - 1}{1}$

Step 2.3.2.4

Divide $a - 1$ by $1$ .

$a - 1$