Divide Using Synthetic Division (10x^3+x^2-21x+9)/(5x-7)

$\frac{10 x^{3} + x^{2} - 21 x + 9}{5 x - 7}$

Step 1

Divide each term in the denominator by $5$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{5} \cdot \frac{10 x^{3} + x^{2} - 21 x + 9}{\frac{5 x - 7}{5}}$

Step 2

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

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0			\xa0

Step 3

The first number in the dividend $(10)$ is put into the first position of the result area (below the horizontal line).

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0			\xa0
	$10$

Step 4

Multiply the newest entry in the result $(10)$ by the divisor $(\frac{7}{5})$ and place the result of $(14)$ under the next term in the dividend $(1)$ .

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$		\xa0
	$10$

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.

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$		\xa0
	$10$	$15$

Step 6

Multiply the newest entry in the result $(15)$ by the divisor $(\frac{7}{5})$ and place the result of $(21)$ under the next term in the dividend $(- 21)$ .

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$	$21$	\xa0
	$10$	$15$

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.

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$	$21$	\xa0
	$10$	$15$	$0$

Step 8

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{7}{5})$ and place the result of $(0)$ under the next term in the dividend $(9)$ .

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$	$21$	$0$
	$10$	$15$	$0$

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.

$\frac{7}{5}$	$10$	$1$	$- 21$	$9$
	\xa0	$14$	$21$	$0$
	$10$	$15$	$0$	$9$

Step 10

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

$(\frac{1}{5}) \cdot (10 x^{2} + 15 x + 0 + \frac{9}{\frac{5 x - 7}{5}})$

Step 11

Simplify the quotient polynomial.

$(\frac{1}{5}) \cdot (10 x^{2} + 15 x + \frac{45}{5 x - 7})$

Step 12

Simplify.

Distribute.

Apply the distributive property.

$\frac{1}{5} \cdot (10 x^{2} + 15 x) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Apply the distributive property.

$\frac{1}{5} \cdot (10 x^{2}) + \frac{1}{5} \cdot (15 x) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Cancel the common factor of $5$ .

Factor $5$ out of $10 x^{2}$ .

$\frac{1}{5} \cdot (5 (2 x^{2})) + \frac{1}{5} \cdot (15 x) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Cancel the common factor.

$\frac{1}{5} \cdot (5 (2 x^{2})) + \frac{1}{5} \cdot (15 x) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Rewrite the expression.

$2 x^{2} + \frac{1}{5} \cdot (15 x) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Cancel the common factor of $5$ .

Factor $5$ out of $15 x$ .

$2 x^{2} + \frac{1}{5} \cdot (5 (3 x)) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Cancel the common factor.

$2 x^{2} + \frac{1}{5} \cdot (5 (3 x)) + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Rewrite the expression.

$2 x^{2} + 3 x + \frac{1}{5} \cdot \frac{45}{5 x - 7}$

Cancel the common factor of $5$ .

Factor $5$ out of $45$ .

$2 x^{2} + 3 x + \frac{1}{5} \cdot \frac{5 (9)}{5 x - 7}$

Cancel the common factor.

$2 x^{2} + 3 x + \frac{1}{5} \cdot \frac{5 \cdot 9}{5 x - 7}$

Rewrite the expression.

$2 x^{2} + 3 x + \frac{9}{5 x - 7}$

Divide Using Synthetic Division (10x^3+x^2-21x+9)/(5x-7)

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