use synthetic division to solve $(x^{3}+1)div(x - 1)$. what is the quotient?\n$x^{2}+x + 1+\frac{2}{x +…

Explanation:

Step1: Set up synthetic division

The divisor is $x - 1$, so we use $c = 1$. The dividend is $x^{3}+0x^{2}+0x + 1$, and we set up the synthetic - division as follows:

1	1	0	0	1

Step2: Bring down the first coefficient

Bring down the first coefficient 1:

1	1	0	0	1
	1

Step3: Multiply and add

Multiply 1 (the number we brought down) by 1 (the value of $c$) to get 1, and add to the next coefficient 0: $0 + 1=1$.

1	1	0	0	1
		1
	1	1

Multiply 1 by 1 again to get 1, and add to the next coefficient 0: $0 + 1 = 1$.

1	1	0	0	1
		1	1
	1	1	1

Multiply 1 by 1 one more time to get 1, and add to the last coefficient 1: $1+1 = 2$.

1	1	0	0	1
		1	1	1
	1	1	1	2

The numbers 1, 1, 1 are the coefficients of the quotient polynomial, and 2 is the remainder. The quotient is $x^{2}+x + 1$ and the remainder is 2. So, $\frac{x^{3}+1}{x - 1}=x^{2}+x + 1+\frac{2}{x - 1}$.

Answer:

$x^{2}+x + 1+\frac{2}{x - 1}$