To divide $x^3 + x^2 - 4x + 3$ by $x - 2$ using synthetic division, we follow these steps:

1. Write down the coefficients of $x^3 + x^2 - 4x + 3$: $1, 1, -4, 3$.
2. Since we're dividing by $x - 2$, the number we use in synthetic division is $2$.

Let's proceed with the synthetic division:

Synthetic Division Steps:

1. Write down the coefficients: $1, 1, -4, 3$.
2. Bring down the first coefficient $1$ as it is.
3. Multiply this $1$ by $2$ and write the result ($2$) below the next coefficient:
   • The second column becomes $1 + 2 = 3$.
4. Multiply $3$ by $2$ and write the result ($6$) below the next coefficient:
   • The third column becomes $-4 + 6 = 2$.
5. Multiply $2$ by $2$ and write the result ($4$) below the last coefficient:
   • The fourth column becomes $3 + 4 = 7$.

Result:

The bottom row represents the coefficients of the quotient and the remainder:

• Quotient: $x^2 + 3x + 2$
• Remainder: $7$

So, the answer is:

• Quotient: $x^2 + 3x + 2$
• Remainder: $7$

Would you like further details on this process or have any questions?

Related Questions:

1. How do you apply synthetic division to higher-degree polynomials?
2. What happens if the divisor is in a different form, such as $x + a$?
3. How can synthetic division help find factors of a polynomial?
4. How is synthetic division different from long division of polynomials?
5. How does the remainder theorem relate to synthetic division?

Tip:

When using synthetic division, always ensure to bring down the first coefficient directly and multiply consistently to avoid mistakes.