Pre-Algebra Examples

$\frac{a^{3} + a^{2} - 4 a - 4}{a - 2}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$\frac{(a^{3} + a^{2}) - 4 a - 4}{a - 2}$

Step 1.1.2

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

$\frac{a^{2} (a + 1) - 4 (a + 1)}{a - 2}$

Step 1.2

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

$\frac{(a + 1) (a^{2} - 4)}{a - 2}$

Step 1.3

Rewrite $4$ as $2^{2}$ .

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

Step 1.4

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

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

Step 2

Cancel the common factor of $a - 2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(a + 1) (a + 2)$ by $1$ .

$(a + 1) (a + 2)$

Step 3

Expand $(a + 1) (a + 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$a (a + 2) + 1 (a + 2)$

Step 3.2

Apply the distributive property.

$a \cdot a + a \cdot 2 + 1 (a + 2)$

Step 3.3

Apply the distributive property.

$a \cdot a + a \cdot 2 + 1 a + 1 \cdot 2$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $a$ by $a$ .

$a^{2} + a \cdot 2 + 1 a + 1 \cdot 2$

Step 4.1.2

Move $2$ to the left of $a$ .

$a^{2} + 2 \cdot a + 1 a + 1 \cdot 2$

Step 4.1.3

Multiply $a$ by $1$ .

$a^{2} + 2 a + a + 1 \cdot 2$

Step 4.1.4

Multiply $2$ by $1$ .

$a^{2} + 2 a + a + 2$

Step 4.2

Add $2 a$ and $a$ .

$a^{2} + 3 a + 2$