Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $6 x^{2}$ out of $6 x^{4} - 12 x^{3} + 6 x^{2}$ .

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Step 1.1.1

Factor $6 x^{2}$ out of $6 x^{4}$ .

$\frac{6 x^{2} (x^{2}) - 12 x^{3} + 6 x^{2}}{3 x^{2}}$

Step 1.1.2

Factor $6 x^{2}$ out of $- 12 x^{3}$ .

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

Step 1.1.3

Factor $6 x^{2}$ out of $6 x^{2}$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

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

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.

$2 x = 2 \cdot x \cdot 1$

Step 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

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

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

Step 2

Cancel the common factor of $6$ and $3$ .

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Step 2.1

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{2}$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $2 {(x - 1)}^{2}$ by $1$ .

$2 {(x - 1)}^{2}$

Step 4

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 1$ to the left of $x$ .

$2 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Step 6.1.3

Rewrite $- 1 x$ as $- x$ .

$2 (x^{2} - x - 1 x - 1 \cdot - 1)$

Step 6.1.4

Rewrite $- 1 x$ as $- x$ .

$2 (x^{2} - x - x - 1 \cdot - 1)$

Step 6.1.5

Multiply $- 1$ by $- 1$ .

$2 (x^{2} - x - x + 1)$

Step 6.2

Subtract $x$ from $- x$ .

$2 (x^{2} - 2 x + 1)$

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Multiply $- 2$ by $2$ .

$2 x^{2} - 4 x + 2 \cdot 1$

Step 8.2

Multiply $2$ by $1$ .

$2 x^{2} - 4 x + 2$