Calculus Examples

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

Step 1

Divide each term in the denominator by $2$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{2} \cdot \frac{4 x^{3} - 2 x^{2} - 8 x + 4}{x - 1}$

Step 2

Place the numbers representing the divisor and the dividend into a division-like configuration.

$1$	$4$	$- 2$	$- 8$	$4$

Step 3

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$1$	$4$	$- 2$	$- 8$	$4$

	$4$

Step 4

Multiply the newest entry in the result $(4)$ by the divisor $(1)$ and place the result of $(4)$ under the next term in the dividend $(- 2)$ .

$1$	$4$	$- 2$	$- 8$	$4$
		$4$
	$4$

Step 5

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$4$	$- 2$	$- 8$	$4$
		$4$
	$4$	$2$

Step 6

Multiply the newest entry in the result $(2)$ by the divisor $(1)$ and place the result of $(2)$ under the next term in the dividend $(- 8)$ .

$1$	$4$	$- 2$	$- 8$	$4$
		$4$	$2$
	$4$	$2$

Step 7

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$4$	$- 2$	$- 8$	$4$
		$4$	$2$
	$4$	$2$	$- 6$

Step 8

Multiply the newest entry in the result $(- 6)$ by the divisor $(1)$ and place the result of $(- 6)$ under the next term in the dividend $(4)$ .

$1$	$4$	$- 2$	$- 8$	$4$
		$4$	$2$	$- 6$
	$4$	$2$	$- 6$

Step 9

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$4$	$- 2$	$- 8$	$4$
		$4$	$2$	$- 6$
	$4$	$2$	$- 6$	$- 2$

Step 10

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$(\frac{1}{2}) \cdot (4 x^{2} + 2 x - 6 + \frac{- 2}{x - 1})$

Step 11

Simplify the quotient polynomial.

$(\frac{1}{2}) \cdot (4 x^{2} + 2 x - 6 - \frac{2}{x - 1})$

Step 12

Simplify.

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Step 12.1

Distribute.

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Step 12.1.1

Apply the distributive property.

$\frac{1}{2} \cdot (4 x^{2} + 2 x - 6) + \frac{1}{2} \cdot (- \frac{2}{x - 1})$

Step 12.1.2

Apply the distributive property.

$\frac{1}{2} \cdot (4 x^{2} + 2 x) + \frac{1}{2} \cdot - 6 + \frac{1}{2} \cdot (- \frac{2}{x - 1})$

Step 12.1.3

Apply the distributive property.

$\frac{1}{2} \cdot (4 x^{2}) + \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot - 6 + \frac{1}{2} \cdot (- \frac{2}{x - 1})$

Step 12.2

Cancel the common factor of $2$ .

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Step 12.2.1

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

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

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

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

Step 12.3

Cancel the common factor of $2$ .

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Step 12.3.1

Factor $2$ out of $2 x$ .

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

Step 12.3.2

Cancel the common factor.

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

Step 12.3.3

Rewrite the expression.

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

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Factor $2$ out of $- 6$ .

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

Step 12.4.2

Cancel the common factor.

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

Step 12.4.3

Rewrite the expression.

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

Step 12.5

Cancel the common factor of $2$ .

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Step 12.5.1

Move the leading negative in $- \frac{2}{x - 1}$ into the numerator.

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

Step 12.5.2

Factor $2$ out of $- 2$ .

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

Step 12.5.3

Cancel the common factor.

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

Step 12.5.4

Rewrite the expression.

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

Step 12.6

Move the negative in front of the fraction.

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