Questions: Use synthetic division to divide, where the result involves a complex number. Specify the quotient and the remainder. (x^3 + 1) / (x - 1) quotient remainder Recall that only the coefficients of the terms of the polynomial listed in descending powers of the variable are used in synthetic

Solution Steps

To solve the problem using synthetic division, we need to follow these steps:

1. Identify the coefficients of the polynomial $x^3 + 1$.
2. Use the root of the divisor $x - 1$, which is $1$, for synthetic division.
3. Perform the synthetic division process to find the quotient and the remainder.

The polynomial $x^3 + 1$ can be expressed in terms of its coefficients as follows: $$\text{Coefficients} = [1, 0, 0, 1]$$ This corresponds to $1x^3 + 0x^2 + 0x + 1$.

Using the root of the divisor $x - 1$, which is $1$, we perform synthetic division. The process yields the following results:

• The first coefficient is $1$.
• The next coefficients are calculated as follows:
  • $0 + 1 \cdot 1 = 1$
  • $0 + 1 \cdot 1 = 1$
  • $1 + 1 \cdot 1 = 2$

Thus, the synthetic division results in: $$\text{Quotient} = [1, 1, 2]$$ and the remainder is: $$\text{Remainder} = 2$$

The quotient can be expressed as: $$x^2 + x + 2$$ and the remainder is: $$2$$

Final Answer