Examples

f (x) = x^{3} - 7 x

$f (x) = x^{3} - 7 x$ ,

x = 0

$x = 0$

Step 1

Set up the long division problem to evaluate the function at

0

$0$ .

\frac{x^{3} - 7 x}{x - (0)}

$\frac{x^{3} - 7 x}{x - (0)}$

Step 2

Divide using synthetic division.

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Step 2.1

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

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$

Step 2.2

The first number in the dividend

(1)

$(1)$ is put into the first position of the result area (below the horizontal line).

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$

	$1$ $1$

Step 2.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(0)

$(0)$ and place the result of

(0)

$(0)$ under the next term in the dividend

(0)

$(0)$ .

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$
		$0$ $0$
	$1$ $1$

Step 2.4

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

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$
		$0$ $0$
	$1$ $1$	$0$ $0$

Step 2.5

Multiply the newest entry in the result

(0)

$(0)$ by the divisor

(0)

$(0)$ and place the result of

(0)

$(0)$ under the next term in the dividend

(- 7)

$(- 7)$ .

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$
		$0$ $0$	$0$ $0$
	$1$ $1$	$0$ $0$

Step 2.6

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

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$
		$0$ $0$	$0$ $0$
	$1$ $1$	$0$ $0$	$- 7$ $- 7$

Step 2.7

Multiply the newest entry in the result

(- 7)

$(- 7)$ by the divisor

(0)

$(0)$ and place the result of

(0)

$(0)$ under the next term in the dividend

(0)

$(0)$ .

$0$ $0$	$1$ $1$	$0$ $0$	$- 7$ $- 7$	$0$ $0$
		$0$ $0$	$0$ $0$	$0$ $0$
	$1$ $1$	$0$ $0$	$- 7$ $- 7$

Step 2.8

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

$0$ $0$	$1$ $1$	$0$	$- 7$	$0$
		$0$	$0$	$0$
	$1$	$0$	$- 7$	$0$

Step 2.9

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

$1 x^{2} + 0 x - 7$

Step 2.10

Simplify the quotient polynomial.

$x^{2} - 7$

Step 3

The remainder of the synthetic division is the result based on the remainder theorem.

$0$

Step 4

Since the remainder is equal to zero,

$x = 0$ is a factor.

$x = 0$ is a factor

Step 5

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