Finite Math Examples

$\frac{4 x^{3} - 12 x^{2} - x + 15}{3 x - 4}$

Step 1

Divide each term in the denominator by $3$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{3} \cdot \frac{4 x^{3} - 12 x^{2} - x + 15}{\frac{3 x - 4}{3}}$

Step 2

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

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$

Step 3

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

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 4

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{4}{3})$ and place the result of $(\frac{16}{3})$ under the next term in the dividend $(- 12)$ .

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$
	$4$

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.

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$
	$4$	$- \frac{20}{3}$

Step 6

Multiply the newest entry in the result $(- \frac{20}{3})$ by the divisor $(\frac{4}{3})$ and place the result of $(- \frac{80}{9})$ under the next term in the dividend $(- 1)$ .

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$	$- \frac{80}{9}$
	$4$	$- \frac{20}{3}$

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.

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$	$- \frac{80}{9}$
	$4$	$- \frac{20}{3}$	$- \frac{89}{9}$

Step 8

Multiply the newest entry in the result $(- \frac{89}{9})$ by the divisor $(\frac{4}{3})$ and place the result of $(- \frac{356}{27})$ under the next term in the dividend $(15)$ .

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$	$- \frac{80}{9}$	$- \frac{356}{27}$
	$4$	$- \frac{20}{3}$	$- \frac{89}{9}$

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.

$\frac{4}{3}$	$4$	$- 12$	$- 1$	$15$
		$\frac{16}{3}$	$- \frac{80}{9}$	$- \frac{356}{27}$
	$4$	$- \frac{20}{3}$	$- \frac{89}{9}$	$\frac{49}{27}$

Step 10

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

$(\frac{1}{3}) \cdot (4 x^{2} + - \frac{20}{3} x - \frac{89}{9} + \frac{\frac{49}{27}}{\frac{3 x - 4}{3}})$

Step 11

Simplify the quotient polynomial.

$(\frac{1}{3}) \cdot (4 x^{2} - \frac{20 x}{3} - \frac{89}{9} + \frac{49}{9 (3 x - 4)})$

Step 12

Simplify.

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Step 12.1

Distribute.

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Step 12.1.1

Apply the distributive property.

$\frac{1}{3} \cdot (4 x^{2} - \frac{20 x}{3} - \frac{89}{9} + \frac{49}{9 (3 x) + 9 \cdot - 4})$

Step 12.1.2

Apply the distributive property.

$\frac{1}{3} \cdot (4 x^{2} - \frac{20 x}{3} - \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.1.3

Apply the distributive property.

$\frac{1}{3} \cdot (4 x^{2} - \frac{20 x}{3}) + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.1.4

Apply the distributive property.

$\frac{1}{3} \cdot (4 x^{2}) + \frac{1}{3} \cdot (- \frac{20 x}{3}) + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.2

Multiply $\frac{1}{3} (4 x^{2})$ .

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Step 12.2.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} x^{2} + \frac{1}{3} \cdot (- \frac{20 x}{3}) + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.2.2

Combine $\frac{4}{3}$ and $x^{2}$ .

$\frac{4 x^{2}}{3} + \frac{1}{3} \cdot (- \frac{20 x}{3}) + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.3

Multiply $\frac{1}{3} (- \frac{20 x}{3})$ .

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Step 12.3.1

Multiply $\frac{1}{3}$ by $\frac{20 x}{3}$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{3 \cdot 3} + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.3.2

Multiply $3$ by $3$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} + \frac{1}{3} \cdot (- \frac{89}{9}) + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.4

Multiply $\frac{1}{3} (- \frac{89}{9})$ .

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Step 12.4.1

Multiply $\frac{1}{3}$ by $\frac{89}{9}$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{3 \cdot 9} + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.4.2

Multiply $3$ by $9$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 \cdot - 4}$

Step 12.5

Simplify the denominator.

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Step 12.5.1

Multiply $3$ by $9$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{27 x + 9 \cdot - 4}$

Step 12.5.2

Multiply $9$ by $- 4$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{27 x - 36}$

Step 12.5.3

Factor $9$ out of $27 x - 36$ .

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Step 12.5.3.1

Factor $9$ out of $27 x$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{9 (3 x) - 36}$

Step 12.5.3.2

Factor $9$ out of $- 36$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{9 (3 x) + 9 (- 4)}$

Step 12.5.3.3

Factor $9$ out of $9 (3 x) + 9 (- 4)$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{1}{3} \cdot \frac{49}{9 (3 x - 4)}$

Step 12.6

Multiply $\frac{1}{3} \cdot \frac{49}{9 (3 x - 4)}$ .

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Step 12.6.1

Multiply $\frac{1}{3}$ by $\frac{49}{9 (3 x - 4)}$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{49}{3 (9 (3 x - 4))}$

Step 12.6.2

Multiply $9$ by $3$ .

$\frac{4 x^{2}}{3} - \frac{20 x}{9} - \frac{89}{27} + \frac{49}{27 (3 x - 4)}$