Evaluate the Function Using Synthetic Division g(x)=3x^6+3x^4-3x^2+6 , g(0)

$g (x) = 3 x^{6} + 3 x^{4} - 3 x^{2} + 6$ , $g (0)$

Step 1

Set up the long division problem to evaluate the function at $0$ .

$\frac{3 x^{6} + 3 x^{4} - 3 x^{2} + 6}{x - (0)}$

Step 2

Divide using synthetic division.

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0						\xa0

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0						\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$					\xa0
	$3$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$					\xa0
	$3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(3)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$				\xa0
	$3$	$0$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$				\xa0
	$3$	$0$	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$			\xa0
	$3$	$0$	$3$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$			\xa0
	$3$	$0$	$3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 3)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$		\xa0
	$3$	$0$	$3$	$0$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$		\xa0
	$3$	$0$	$3$	$0$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$	$0$	\xa0
	$3$	$0$	$3$	$0$	$- 3$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$	$0$	\xa0
	$3$	$0$	$3$	$0$	$- 3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(6)$ .

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$	$0$	$0$
	$3$	$0$	$3$	$0$	$- 3$	$0$

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

$0$	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$
	\xa0	$0$	$0$	$0$	$0$	$0$	$0$
	$3$	$0$	$3$	$0$	$- 3$	$0$	$6$

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

$3 x^{5} + 0 x^{4} + 3 x^{3} + 0 x^{2} + (- 3) x + 0 + \frac{6}{x}$

Simplify the quotient polynomial.

$3 x^{5} + 3 x^{3} - 3 x + \frac{6}{x}$

$3 x^{5} + 3 x^{3} - 3 x + \frac{6}{x}$

Step 3

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

$6$

Step 4

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