Precalculus Examples

$\frac{4 x^{2} + 9 x - 6}{4 x - 3}$

Step 1

Divide each term in the denominator by $4$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{4} \cdot \frac{4 x^{2} + 9 x - 6}{\frac{4 x - 3}{4}}$

Step 2

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

$\frac{3}{4}$	$4$	$9$	$- 6$

Step 3

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

$\frac{3}{4}$	$4$	$9$	$- 6$

	$4$

Step 4

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

$\frac{3}{4}$	$4$	$9$	$- 6$
		$3$
	$4$

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.

$\frac{3}{4}$	$4$	$9$	$- 6$
		$3$
	$4$	$12$

Step 6

Multiply the newest entry in the result $(12)$ by the divisor $(\frac{3}{4})$ and place the result of $(9)$ under the next term in the dividend $(- 6)$ .

$\frac{3}{4}$	$4$	$9$	$- 6$
		$3$	$9$
	$4$	$12$

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.

$\frac{3}{4}$	$4$	$9$	$- 6$
		$3$	$9$
	$4$	$12$	$3$

Step 8

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

$(\frac{1}{4}) \cdot ((4) x + 12 + \frac{3}{\frac{4 x - 3}{4}})$

Step 9

Simplify the quotient polynomial.

$(\frac{1}{4}) \cdot (4 x + 12 + \frac{12}{4 x - 3})$

Step 10

Simplify.

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Step 10.1

Distribute.

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Step 10.1.1

Apply the distributive property.

$\frac{1}{4} \cdot (4 x + 12) + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.1.2

Apply the distributive property.

$\frac{1}{4} \cdot (4 x) + \frac{1}{4} \cdot 12 + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.2

Cancel the common factor of $4$ .

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Step 10.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} \cdot (4 (x)) + \frac{1}{4} \cdot 12 + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.2.2

Cancel the common factor.

$\frac{1}{4} \cdot (4 x) + \frac{1}{4} \cdot 12 + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.2.3

Rewrite the expression.

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

Step 10.3

Cancel the common factor of $4$ .

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Step 10.3.1

Factor $4$ out of $12$ .

$x + \frac{1}{4} \cdot (4 (3)) + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.3.2

Cancel the common factor.

$x + \frac{1}{4} \cdot (4 \cdot 3) + \frac{1}{4} \cdot \frac{12}{4 x - 3}$

Step 10.3.3

Rewrite the expression.

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

Step 10.4

Cancel the common factor of $4$ .

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Step 10.4.1

Factor $4$ out of $12$ .

$x + 3 + \frac{1}{4} \cdot \frac{4 (3)}{4 x - 3}$

Step 10.4.2

Cancel the common factor.

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

Step 10.4.3

Rewrite the expression.

$x + 3 + \frac{3}{4 x - 3}$