Pre-Algebra Examples

$\frac{12 x^{5} - 18 x^{4} + 6 x^{3}}{2 x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $6 x^{3}$ out of $12 x^{5} - 18 x^{4} + 6 x^{3}$ .

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Step 1.1.1

Factor $6 x^{3}$ out of $12 x^{5}$ .

$\frac{6 x^{3} (2 x^{2}) - 18 x^{4} + 6 x^{3}}{2 x^{3}}$

Step 1.1.2

Factor $6 x^{3}$ out of $- 18 x^{4}$ .

$\frac{6 x^{3} (2 x^{2}) + 6 x^{3} (- 3 x) + 6 x^{3}}{2 x^{3}}$

Step 1.1.3

Factor $6 x^{3}$ out of $6 x^{3}$ .

$\frac{6 x^{3} (2 x^{2}) + 6 x^{3} (- 3 x) + 6 x^{3} (1)}{2 x^{3}}$

Step 1.1.4

Factor $6 x^{3}$ out of $6 x^{3} (2 x^{2}) + 6 x^{3} (- 3 x)$ .

$\frac{6 x^{3} (2 x^{2} - 3 x) + 6 x^{3} (1)}{2 x^{3}}$

Step 1.1.5

Factor $6 x^{3}$ out of $6 x^{3} (2 x^{2} - 3 x) + 6 x^{3} (1)$ .

$\frac{6 x^{3} (2 x^{2} - 3 x + 1)}{2 x^{3}}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Step 1.2.1.1

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

$\frac{6 x^{3} (2 x^{2} - 3 (x) + 1)}{2 x^{3}}$

Step 1.2.1.2

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

$\frac{6 x^{3} (2 x^{2} + (- 1 - 2) x + 1)}{2 x^{3}}$

Step 1.2.1.3

Apply the distributive property.

$\frac{6 x^{3} (2 x^{2} - 1 x - 2 x + 1)}{2 x^{3}}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{6 x^{3} ((2 x^{2} - 1 x) - 2 x + 1)}{2 x^{3}}$

Step 1.2.2.2

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

$\frac{6 x^{3} (x (2 x - 1) - (2 x - 1))}{2 x^{3}}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{6 x^{3} ((2 x - 1) (x - 1))}{2 x^{3}}$

$\frac{6 x^{3} (2 x - 1) (x - 1)}{2 x^{3}}$

Step 2

Cancel the common factor of $6$ and $2$ .

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Step 2.1

Factor $2$ out of $6 x^{3} (2 x - 1) (x - 1)$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{3}$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $(3 (2 x - 1)) (x - 1)$ by $1$ .

$(3 (2 x - 1)) (x - 1)$

Step 4

Apply the distributive property.

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

Step 5

Multiply $2$ by $3$ .

$(6 x + 3 \cdot - 1) (x - 1)$

Step 6

Multiply $3$ by $- 1$ .

$(6 x - 3) (x - 1)$

Step 7

Expand $(6 x - 3) (x - 1)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$6 x (x - 1) - 3 (x - 1)$

Step 7.2

Apply the distributive property.

$6 x \cdot x + 6 x \cdot - 1 - 3 (x - 1)$

Step 7.3

Apply the distributive property.

$6 x \cdot x + 6 x \cdot - 1 - 3 x - 3 \cdot - 1$

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 $x$ by $x$ by adding the exponents.

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Step 8.1.1.1

Move $x$ .

$6 (x \cdot x) + 6 x \cdot - 1 - 3 x - 3 \cdot - 1$

Step 8.1.1.2

Multiply $x$ by $x$ .

$6 x^{2} + 6 x \cdot - 1 - 3 x - 3 \cdot - 1$

Step 8.1.2

Multiply $- 1$ by $6$ .

$6 x^{2} - 6 x - 3 x - 3 \cdot - 1$

Step 8.1.3

Multiply $- 3$ by $- 1$ .

$6 x^{2} - 6 x - 3 x + 3$

Step 8.2

Subtract $3 x$ from $- 6 x$ .

$6 x^{2} - 9 x + 3$