To solve this problem using synthetic division, let’s go through the steps in detail.

Given Polynomial

The polynomial is: $x^4 - 3x^3 + 2x^2 + 4x + 4$ and we need to divide it by $x - 2$.

Synthetic Division Setup

1. Write down the coefficients of the polynomial: $1, -3, 2, 4, 4$.
2. Since we are dividing by $x - 2$, the value we'll use in synthetic division is $2$.

Synthetic Division Steps

1. Bring down the first coefficient (1).
2. Multiply $2$ by the current result, then add it to the next coefficient.

Let’s go through each step:

Final Row (Quotient and Remainder)

The bottom row gives us the coefficients of the quotient and the remainder: $1, -1, 0, 4 \quad \text{with remainder } 12$

Answer

• Quotient: $x^3 - x^2 + 4$
• Remainder: $12$

So, the answer is: $\text{Quotient is } x^3 - x^2 + 4, \text{ and the remainder is } 12.$

Would you like more details on each synthetic division step or have any questions?

Related Questions

1. How do you use synthetic division for polynomials of lower degrees?
2. What is the relationship between the remainder in synthetic division and the Remainder Theorem?
3. Can synthetic division be used to divide by other terms, like $x + 3$?
4. How can synthetic division help identify factors of a polynomial?
5. How would the result change if we divided by $x + 2$ instead of $x - 2$?

Tip

Synthetic division is only applicable when dividing by linear divisors of the form $x - c$.