Calculus Examples

$5 x - 10 \div x^{2} - 4 x + 4 + 1 \div (x - 2)$

Step 1

Simplify each term.

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Step 1.1

Rewrite the division as a fraction.

$5 x - \frac{10}{x^{2}} - 4 x + 4 + 1 \div (x - 2)$

Step 1.2

Rewrite the division as a fraction.

$5 x - \frac{10}{x^{2}} - 4 x + 4 + \frac{1}{x - 2}$

Step 2

Subtract $4 x$ from $5 x$ .

$x - \frac{10}{x^{2}} + 4 + \frac{1}{x - 2}$

Step 3

To write $x$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$- \frac{10}{x^{2}} + 4 + \frac{x (x - 2)}{x - 2} + \frac{1}{x - 2}$

Step 4

Combine the numerators over the common denominator.

$- \frac{10}{x^{2}} + 4 + \frac{x (x - 2) + 1}{x - 2}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$- \frac{10}{x^{2}} + 4 + \frac{x \cdot x + x \cdot - 2 + 1}{x - 2}$

Step 5.2

Multiply $x$ by $x$ .

$- \frac{10}{x^{2}} + 4 + \frac{x^{2} + x \cdot - 2 + 1}{x - 2}$

Step 5.3

Move $- 2$ to the left of $x$ .

$- \frac{10}{x^{2}} + 4 + \frac{x^{2} - 2 \cdot x + 1}{x - 2}$

Step 5.4

Factor using the perfect square rule.

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Step 5.4.1

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

$- \frac{10}{x^{2}} + 4 + \frac{x^{2} - 2 x + 1^{2}}{x - 2}$

Step 5.4.2

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 5.4.3

Rewrite the polynomial.

$- \frac{10}{x^{2}} + 4 + \frac{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}{x - 2}$

Step 5.4.4

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

$- \frac{10}{x^{2}} + 4 + \frac{{(x - 1)}^{2}}{x - 2}$

Step 6

To write $- \frac{10}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$- \frac{10}{x^{2}} \cdot \frac{x - 2}{x - 2} + \frac{{(x - 1)}^{2}}{x - 2} + 4$

Step 7

To write $\frac{{(x - 1)}^{2}}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$- \frac{10}{x^{2}} \cdot \frac{x - 2}{x - 2} + \frac{{(x - 1)}^{2}}{x - 2} \cdot \frac{x^{2}}{x^{2}} + 4$

Step 8

Write each expression with a common denominator of $(x - 2) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{10}{x^{2}}$ by $\frac{x - 2}{x - 2}$ .

$- \frac{10 (x - 2)}{x^{2} (x - 2)} + \frac{{(x - 1)}^{2}}{x - 2} \cdot \frac{x^{2}}{x^{2}} + 4$

Step 8.2

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

$- \frac{10 (x - 2)}{x^{2} (x - 2)} + \frac{{(x - 1)}^{2} x^{2}}{(x - 2) x^{2}} + 4$

Step 8.3

Reorder the factors of $(x - 2) x^{2}$ .

$- \frac{10 (x - 2)}{x^{2} (x - 2)} + \frac{{(x - 1)}^{2} x^{2}}{x^{2} (x - 2)} + 4$

Step 9

Combine the numerators over the common denominator.

$\frac{- 10 (x - 2) + {(x - 1)}^{2} x^{2}}{x^{2} (x - 2)} + 4$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{- 10 x - 10 \cdot - 2 + {(x - 1)}^{2} x^{2}}{x^{2} (x - 2)} + 4$

Step 10.2

Multiply $- 10$ by $- 2$ .

$\frac{- 10 x + 20 + {(x - 1)}^{2} x^{2}}{x^{2} (x - 2)} + 4$

Step 10.3

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

$\frac{- 10 x + 20 + (x - 1) (x - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.4

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

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Step 10.4.1

Apply the distributive property.

$\frac{- 10 x + 20 + (x (x - 1) - 1 (x - 1)) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.4.2

Apply the distributive property.

$\frac{- 10 x + 20 + (x \cdot x + x \cdot - 1 - 1 (x - 1)) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.4.3

Apply the distributive property.

$\frac{- 10 x + 20 + (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5

Simplify and combine like terms.

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Step 10.5.1

Simplify each term.

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Step 10.5.1.1

Multiply $x$ by $x$ .

$\frac{- 10 x + 20 + (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5.1.2

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

$\frac{- 10 x + 20 + (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5.1.3

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

$\frac{- 10 x + 20 + (x^{2} - x - 1 x - 1 \cdot - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5.1.4

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

$\frac{- 10 x + 20 + (x^{2} - x - x - 1 \cdot - 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5.1.5

Multiply $- 1$ by $- 1$ .

$\frac{- 10 x + 20 + (x^{2} - x - x + 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.5.2

Subtract $x$ from $- x$ .

$\frac{- 10 x + 20 + (x^{2} - 2 x + 1) x^{2}}{x^{2} (x - 2)} + 4$

Step 10.6

Apply the distributive property.

$\frac{- 10 x + 20 + x^{2} x^{2} - 2 x \cdot x^{2} + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7

Simplify.

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Step 10.7.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 10.7.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 10 x + 20 + x^{2 + 2} - 2 x \cdot x^{2} + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.1.2

Add $2$ and $2$ .

$\frac{- 10 x + 20 + x^{4} - 2 x \cdot x^{2} + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 10.7.2.1

Move $x^{2}$ .

$\frac{- 10 x + 20 + x^{4} - 2 (x^{2} x) + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.2.2

Multiply $x^{2}$ by $x$ .

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Step 10.7.2.2.1

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

$\frac{- 10 x + 20 + x^{4} - 2 (x^{2} x^{1}) + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 10 x + 20 + x^{4} - 2 x^{2 + 1} + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.2.3

Add $2$ and $1$ .

$\frac{- 10 x + 20 + x^{4} - 2 x^{3} + 1 x^{2}}{x^{2} (x - 2)} + 4$

Step 10.7.3

Multiply $x^{2}$ by $1$ .

$\frac{- 10 x + 20 + x^{4} - 2 x^{3} + x^{2}}{x^{2} (x - 2)} + 4$

Step 10.8

Reorder terms.

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20}{x^{2} (x - 2)} + 4$

Step 11

To write $4$ as a fraction with a common denominator, multiply by $\frac{(x - 2) x^{2}}{(x - 2) x^{2}}$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20}{x^{2} (x - 2)} + 4 \cdot \frac{(x - 2) x^{2}}{(x - 2) x^{2}}$

Step 12

Write each expression with a common denominator of $(x - 2) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.1

Combine $4$ and $\frac{(x - 2) x^{2}}{(x - 2) x^{2}}$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20}{x^{2} (x - 2)} + \frac{4 ((x - 2) x^{2})}{(x - 2) x^{2}}$

Step 12.2

Reorder the factors of $(x - 2) x^{2}$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20}{x^{2} (x - 2)} + \frac{4 ((x - 2) x^{2})}{x^{2} (x - 2)}$

Step 13

Combine the numerators over the common denominator.

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 ((x - 2) x^{2})}{x^{2} (x - 2)}$

Step 14

Simplify the numerator.

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Step 14.1

Apply the distributive property.

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + (4 x + 4 \cdot - 2) x^{2}}{x^{2} (x - 2)}$

Step 14.2

Multiply $4$ by $- 2$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + (4 x - 8) x^{2}}{x^{2} (x - 2)}$

Step 14.3

Apply the distributive property.

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 x \cdot x^{2} - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 14.4.1

Move $x^{2}$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 (x^{2} x) - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.4.2

Multiply $x^{2}$ by $x$ .

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Step 14.4.2.1

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

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 (x^{2} x^{1}) - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 x^{2 + 1} - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.4.3

Add $2$ and $1$ .

$\frac{x^{4} - 2 x^{3} + x^{2} - 10 x + 20 + 4 x^{3} - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.5

Add $- 2 x^{3}$ and $4 x^{3}$ .

$\frac{x^{4} + 2 x^{3} + x^{2} - 10 x + 20 - 8 x^{2}}{x^{2} (x - 2)}$

Step 14.6

Subtract $8 x^{2}$ from $x^{2}$ .

$\frac{x^{4} + 2 x^{3} - 7 x^{2} - 10 x + 20}{x^{2} (x - 2)}$