Answer :
- Bring down the first coefficient.
- Multiply the divisor by the current result and add it to the next coefficient.
- Repeat the multiplication and addition steps until all coefficients have been processed.
- The final result is the remainder: $\boxed{{-17}}$.
### Explanation
1. Understanding the Problem
We are given a synthetic division problem and asked to find the remainder. The problem is set up as follows:
$2 \overline{) 2 \quad -4 \quad -9 \quad 1}$
This represents dividing the polynomial $2x^3 - 4x^2 - 9x + 1$ by $x-2$. We will perform synthetic division to find the remainder.
2. Step 1: Bring Down the First Coefficient
First, bring down the leading coefficient (2).
3. Step 2: Multiply and Write
Multiply the divisor (2) by the brought-down coefficient (2) to get 4. Write this under the next coefficient (-4).
4. Step 3: Add
Add -4 and 4 to get 0.
5. Step 4: Multiply and Write
Multiply the divisor (2) by 0 to get 0. Write this under the next coefficient (-9).
6. Step 5: Add
Add -9 and 0 to get -9.
7. Step 6: Multiply and Write
Multiply the divisor (2) by -9 to get -18. Write this under the last coefficient (1).
8. Step 7: Add to Find Remainder
Add 1 and -18 to get -17. This is the remainder.
9. Conclusion
Therefore, the remainder is -17.
### Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - a. It's often used in algebra to find the roots of polynomials or to simplify expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might use polynomial equations. Synthetic division can help simplify these calculations, making the design process more efficient and accurate. It also helps in fields like signal processing where polynomials are used to model systems.
- Multiply the divisor by the current result and add it to the next coefficient.
- Repeat the multiplication and addition steps until all coefficients have been processed.
- The final result is the remainder: $\boxed{{-17}}$.
### Explanation
1. Understanding the Problem
We are given a synthetic division problem and asked to find the remainder. The problem is set up as follows:
$2 \overline{) 2 \quad -4 \quad -9 \quad 1}$
This represents dividing the polynomial $2x^3 - 4x^2 - 9x + 1$ by $x-2$. We will perform synthetic division to find the remainder.
2. Step 1: Bring Down the First Coefficient
First, bring down the leading coefficient (2).
3. Step 2: Multiply and Write
Multiply the divisor (2) by the brought-down coefficient (2) to get 4. Write this under the next coefficient (-4).
4. Step 3: Add
Add -4 and 4 to get 0.
5. Step 4: Multiply and Write
Multiply the divisor (2) by 0 to get 0. Write this under the next coefficient (-9).
6. Step 5: Add
Add -9 and 0 to get -9.
7. Step 6: Multiply and Write
Multiply the divisor (2) by -9 to get -18. Write this under the last coefficient (1).
8. Step 7: Add to Find Remainder
Add 1 and -18 to get -17. This is the remainder.
9. Conclusion
Therefore, the remainder is -17.
### Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - a. It's often used in algebra to find the roots of polynomials or to simplify expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might use polynomial equations. Synthetic division can help simplify these calculations, making the design process more efficient and accurate. It also helps in fields like signal processing where polynomials are used to model systems.
