Examples

f (x) = x^{2} - 5 x + 4

$f (x) = x^{2} - 5 x + 4$ ,

x = - 4

$x = - 4$

Step 1

Set up the long division problem to evaluate the function at

- 4

$- 4$ .

\frac{x^{2} - 5 x + 4}{x - (- 4)}

$\frac{x^{2} - 5 x + 4}{x - (- 4)}$

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.

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$

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).

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$

	$1$ $1$

Step 2.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(- 4)

$(- 4)$ and place the result of

(- 4)

$(- 4)$ under the next term in the dividend

(- 5)

$(- 5)$ .

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$
		$- 4$ $- 4$
	$1$ $1$

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.

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$
		$- 4$ $- 4$
	$1$ $1$	$- 9$ $- 9$

Step 2.5

Multiply the newest entry in the result

(- 9)

$(- 9)$ by the divisor

(- 4)

$(- 4)$ and place the result of

(36)

$(36)$ under the next term in the dividend

(4)

$(4)$ .

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$
		$- 4$ $- 4$	$36$ $36$
	$1$ $1$	$- 9$ $- 9$

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.

$- 4$ $- 4$	$1$ $1$	$- 5$ $- 5$	$4$ $4$
		$- 4$ $- 4$	$36$ $36$
	$1$ $1$	$- 9$ $- 9$	$40$ $40$

Step 2.7

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

(1) x - 9 + \frac{40}{x + 4}

$(1) x - 9 + \frac{40}{x + 4}$

Step 2.8

Simplify the quotient polynomial.

x - 9 + \frac{40}{x + 4}

$x - 9 + \frac{40}{x + 4}$

x - 9 + \frac{40}{x + 4}

$x - 9 + \frac{40}{x + 4}$

Step 3

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

40

$40$

Step 4

Since the remainder is not equal to zero,

x = - 4

$x = - 4$ is not a factor.

x = - 4

$x = - 4$ is not a factor

Step 5

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