Pre-Algebra Examples

$\frac{12 a^{3} - 8 a + 4 a^{2}}{2 a}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4 a$ out of $12 a^{3} - 8 a + 4 a^{2}$ .

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Step 1.1.1

Factor $4 a$ out of $12 a^{3}$ .

$\frac{4 a (3 a^{2}) - 8 a + 4 a^{2}}{2 a}$

Step 1.1.2

Factor $4 a$ out of $- 8 a$ .

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

Step 1.1.3

Factor $4 a$ out of $4 a^{2}$ .

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

Step 1.1.4

Factor $4 a$ out of $4 a (3 a^{2}) + 4 a (- 2)$ .

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

Step 1.1.5

Factor $4 a$ out of $4 a (3 a^{2} - 2) + 4 a (a)$ .

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

Step 1.2

Factor by grouping.

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Step 1.2.1

Reorder terms.

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

Step 1.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Step 1.2.2.1

Multiply by $1$ .

$\frac{4 a (3 a^{2} + 1 a - 2)}{2 a}$

Step 1.2.2.2

Rewrite $1$ as $- 2$ plus $3$

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

Step 1.2.2.3

Apply the distributive property.

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

Step 1.2.3

Factor out the greatest common factor from each group.

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Step 1.2.3.1

Group the first two terms and the last two terms.

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

Step 1.2.3.2

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

$\frac{4 a (a (3 a - 2) + 1 (3 a - 2))}{2 a}$

Step 1.2.4

Factor the polynomial by factoring out the greatest common factor, $3 a - 2$ .

$\frac{4 a ((3 a - 2) (a + 1))}{2 a}$

$\frac{4 a (3 a - 2) (a + 1)}{2 a}$

Step 2

Cancel the common factor of $4$ and $2$ .

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Step 2.1

Factor $2$ out of $4 a (3 a - 2) (a + 1)$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $2 a$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $a$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $(2 (3 a - 2)) (a + 1)$ by $1$ .

$(2 (3 a - 2)) (a + 1)$

Step 4

Apply the distributive property.

$(2 (3 a) + 2 \cdot - 2) (a + 1)$

Step 5

Multiply $3$ by $2$ .

$(6 a + 2 \cdot - 2) (a + 1)$

Step 6

Multiply $2$ by $- 2$ .

$(6 a - 4) (a + 1)$

Step 7

Expand $(6 a - 4) (a + 1)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$6 a (a + 1) - 4 (a + 1)$

Step 7.2

Apply the distributive property.

$6 a \cdot a + 6 a \cdot 1 - 4 (a + 1)$

Step 7.3

Apply the distributive property.

$6 a \cdot a + 6 a \cdot 1 - 4 a - 4 \cdot 1$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 8.1.1.1

Move $a$ .

$6 (a \cdot a) + 6 a \cdot 1 - 4 a - 4 \cdot 1$

Step 8.1.1.2

Multiply $a$ by $a$ .

$6 a^{2} + 6 a \cdot 1 - 4 a - 4 \cdot 1$

Step 8.1.2

Multiply $6$ by $1$ .

$6 a^{2} + 6 a - 4 a - 4 \cdot 1$

Step 8.1.3

Multiply $- 4$ by $1$ .

$6 a^{2} + 6 a - 4 a - 4$

Step 8.2

Subtract $4 a$ from $6 a$ .

$6 a^{2} + 2 a - 4$