Evaluate Using the Remainder Theorem P(x)=-4x^3-x^2+9x-18 , c=9

$P (x) = - 4 x^{3} - x^{2} + 9 x - 18$ , $c = 9$

Step 1

Set up the long division problem to evaluate the function at $9$ .

$\frac{- 4 x^{3} - x^{2} + 9 x - 18}{x - (9)}$

Step 2

Divide using synthetic division.

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

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0			\xa0

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

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0			\xa0
	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(9)$ and place the result of $(- 36)$ under the next term in the dividend $(- 1)$ .

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$		\xa0
	$- 4$

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

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$		\xa0
	$- 4$	$- 37$

Multiply the newest entry in the result $(- 37)$ by the divisor $(9)$ and place the result of $(- 333)$ under the next term in the dividend $(9)$ .

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$	$- 333$	\xa0
	$- 4$	$- 37$

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

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$	$- 333$	\xa0
	$- 4$	$- 37$	$- 324$

Multiply the newest entry in the result $(- 324)$ by the divisor $(9)$ and place the result of $(- 2916)$ under the next term in the dividend $(- 18)$ .

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$	$- 333$	$- 2916$
	$- 4$	$- 37$	$- 324$

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

$9$	$- 4$	$- 1$	$9$	$- 18$
	\xa0	$- 36$	$- 333$	$- 2916$
	$- 4$	$- 37$	$- 324$	$- 2934$

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

$- 4 x^{2} + - 37 x - 324 + \frac{- 2934}{x - 9}$

Simplify the quotient polynomial.

$- 4 x^{2} - 37 x - 324 - \frac{2934}{x - 9}$

Step 3

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

$- 2934$

Step 4

Evaluate Using the Remainder Theorem P(x)=-4x^3-x^2+9x-18 , c=9

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