Answer :
Sure! Let's go through the process of synthetic division step-by-step to find the quotient and remainder when dividing the polynomial [tex]\( 2x^3 + 9x^2 - 4x + 5 \)[/tex] by [tex]\( x + 5 \)[/tex].
### Steps in Synthetic Division:
1. Identify the Root of the Divisor:
- We are dividing by [tex]\( x + 5 \)[/tex], which means we use the root [tex]\( x = -5 \)[/tex] for synthetic division.
2. Set up the Coefficients:
- Write down the coefficients of the polynomial: [tex]\( [2, 9, -4, 5] \)[/tex].
3. Perform the Division:
- Start with the first coefficient. This will be the first number in your row, which is [tex]\( 2 \)[/tex].
4. Calculate the Quotient Coefficients:
- Multiply the root [tex]\(-5\)[/tex] by the first number [tex]\(2\)[/tex] and write the result under the next coefficient.
- Add this result to the next coefficient [tex]\(9\)[/tex] to get the new number.
- Repeat this process across all coefficients.
Here are the detailed steps:
- Step 1:
- Bring down the first coefficient [tex]\(2\)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\( 2 \times (-5) = -10 \)[/tex].
- Step 2:
- Add to the next coefficient [tex]\(9\)[/tex]: [tex]\( 9 + (-10) = -1 \)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\(-1 \times (-5) = 5\)[/tex].
- Step 3:
- Add to the next coefficient [tex]\(-4\)[/tex]: [tex]\( -4 + 5 = 1 \)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\( 1 \times (-5) = -5 \)[/tex].
- Step 4:
- Add to the next coefficient [tex]\(5\)[/tex]: [tex]\( 5 + (-5) = 0 \)[/tex].
5. Interpret the Result:
- The numbers obtained after each addition (except the last one) form the coefficients of the quotient polynomial.
- The last number (after the final addition) is the remainder.
Result:
- The quotient is represented by the coefficients [tex]\( [2, -1, 1] \)[/tex], which means the quotient is [tex]\( 2x^2 - x + 1 \)[/tex].
- The remainder is [tex]\( 0 \)[/tex].
Therefore, the quotient of dividing [tex]\( 2x^3 + 9x^2 - 4x + 5 \)[/tex] by [tex]\( x + 5 \)[/tex] is [tex]\( 2x^2 - x + 1 \)[/tex], and the remainder is [tex]\( 0 \)[/tex].
### Steps in Synthetic Division:
1. Identify the Root of the Divisor:
- We are dividing by [tex]\( x + 5 \)[/tex], which means we use the root [tex]\( x = -5 \)[/tex] for synthetic division.
2. Set up the Coefficients:
- Write down the coefficients of the polynomial: [tex]\( [2, 9, -4, 5] \)[/tex].
3. Perform the Division:
- Start with the first coefficient. This will be the first number in your row, which is [tex]\( 2 \)[/tex].
4. Calculate the Quotient Coefficients:
- Multiply the root [tex]\(-5\)[/tex] by the first number [tex]\(2\)[/tex] and write the result under the next coefficient.
- Add this result to the next coefficient [tex]\(9\)[/tex] to get the new number.
- Repeat this process across all coefficients.
Here are the detailed steps:
- Step 1:
- Bring down the first coefficient [tex]\(2\)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\( 2 \times (-5) = -10 \)[/tex].
- Step 2:
- Add to the next coefficient [tex]\(9\)[/tex]: [tex]\( 9 + (-10) = -1 \)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\(-1 \times (-5) = 5\)[/tex].
- Step 3:
- Add to the next coefficient [tex]\(-4\)[/tex]: [tex]\( -4 + 5 = 1 \)[/tex].
- Multiply by [tex]\(-5\)[/tex]: [tex]\( 1 \times (-5) = -5 \)[/tex].
- Step 4:
- Add to the next coefficient [tex]\(5\)[/tex]: [tex]\( 5 + (-5) = 0 \)[/tex].
5. Interpret the Result:
- The numbers obtained after each addition (except the last one) form the coefficients of the quotient polynomial.
- The last number (after the final addition) is the remainder.
Result:
- The quotient is represented by the coefficients [tex]\( [2, -1, 1] \)[/tex], which means the quotient is [tex]\( 2x^2 - x + 1 \)[/tex].
- The remainder is [tex]\( 0 \)[/tex].
Therefore, the quotient of dividing [tex]\( 2x^3 + 9x^2 - 4x + 5 \)[/tex] by [tex]\( x + 5 \)[/tex] is [tex]\( 2x^2 - x + 1 \)[/tex], and the remainder is [tex]\( 0 \)[/tex].
