Answer :
To find the result of dividing the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] by [tex]\(x + 3\)[/tex] using synthetic division, follow these steps:
1. Set up the synthetic division:
- First, note that the divisor is [tex]\(x + 3\)[/tex], which means we will use [tex]\(-3\)[/tex] (the opposite of the constant term in [tex]\(x + 3\)[/tex]) for synthetic division.
- List the coefficients of the polynomial: [tex]\(1, -5, -23, 5, 6\)[/tex].
2. Perform the synthetic division:
- Write [tex]\(-3\)[/tex] on the left, and write the coefficients of the polynomial in a row.
- Bring down the first coefficient (1) to the bottom row.
- Multiply [tex]\(-3\)[/tex] by this number (1) and write the result under the second coefficient. Add this result to the second coefficient [tex]\(-5\)[/tex] and write the sum in the bottom row.
- Repeat the multiply-and-add process for each pair of numbers in your setup.
Let's see the process step-by-step:
```
-3 | 1 -5 -23 5 6
| -3 24 -3 -6
-----------------------
1 -8 1 2 0
```
Here’s what happens during each step:
- The initial 1 is brought down.
- Multiply 1 by [tex]\(-3\)[/tex] to get [tex]\(-3\)[/tex]. Add [tex]\(-3\)[/tex] to [tex]\(-5\)[/tex] to get [tex]\(-8\)[/tex].
- Multiply [tex]\(-8\)[/tex] by [tex]\(-3\)[/tex] to get 24. Add 24 to [tex]\(-23\)[/tex] to get 1.
- Multiply 1 by [tex]\(-3\)[/tex] to get [tex]\(-3\)[/tex]. Add [tex]\(-3\)[/tex] to 5 to get 2.
- Multiply 2 by [tex]\(-3\)[/tex] to get [tex]\(-6\)[/tex]. Add [tex]\(-6\)[/tex] to 6 to get 0.
3. Interpret the results:
- The numbers at the bottom row (except the last number) represent the coefficients of the quotient polynomial. Thus, the quotient is [tex]\(1x^3 - 8x^2 + 1x + 2\)[/tex], or simply [tex]\(x^3 - 8x^2 + x + 2\)[/tex].
- The last number at the bottom row (0) is the remainder. In this case, the remainder is 0, indicating that [tex]\(x+3\)[/tex] is a factor of the given polynomial.
Therefore, when the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] is divided by [tex]\(x + 3\)[/tex], the quotient is [tex]\(x^3 - 8x^2 + x + 2\)[/tex] with a remainder of 0.
1. Set up the synthetic division:
- First, note that the divisor is [tex]\(x + 3\)[/tex], which means we will use [tex]\(-3\)[/tex] (the opposite of the constant term in [tex]\(x + 3\)[/tex]) for synthetic division.
- List the coefficients of the polynomial: [tex]\(1, -5, -23, 5, 6\)[/tex].
2. Perform the synthetic division:
- Write [tex]\(-3\)[/tex] on the left, and write the coefficients of the polynomial in a row.
- Bring down the first coefficient (1) to the bottom row.
- Multiply [tex]\(-3\)[/tex] by this number (1) and write the result under the second coefficient. Add this result to the second coefficient [tex]\(-5\)[/tex] and write the sum in the bottom row.
- Repeat the multiply-and-add process for each pair of numbers in your setup.
Let's see the process step-by-step:
```
-3 | 1 -5 -23 5 6
| -3 24 -3 -6
-----------------------
1 -8 1 2 0
```
Here’s what happens during each step:
- The initial 1 is brought down.
- Multiply 1 by [tex]\(-3\)[/tex] to get [tex]\(-3\)[/tex]. Add [tex]\(-3\)[/tex] to [tex]\(-5\)[/tex] to get [tex]\(-8\)[/tex].
- Multiply [tex]\(-8\)[/tex] by [tex]\(-3\)[/tex] to get 24. Add 24 to [tex]\(-23\)[/tex] to get 1.
- Multiply 1 by [tex]\(-3\)[/tex] to get [tex]\(-3\)[/tex]. Add [tex]\(-3\)[/tex] to 5 to get 2.
- Multiply 2 by [tex]\(-3\)[/tex] to get [tex]\(-6\)[/tex]. Add [tex]\(-6\)[/tex] to 6 to get 0.
3. Interpret the results:
- The numbers at the bottom row (except the last number) represent the coefficients of the quotient polynomial. Thus, the quotient is [tex]\(1x^3 - 8x^2 + 1x + 2\)[/tex], or simply [tex]\(x^3 - 8x^2 + x + 2\)[/tex].
- The last number at the bottom row (0) is the remainder. In this case, the remainder is 0, indicating that [tex]\(x+3\)[/tex] is a factor of the given polynomial.
Therefore, when the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] is divided by [tex]\(x + 3\)[/tex], the quotient is [tex]\(x^3 - 8x^2 + x + 2\)[/tex] with a remainder of 0.
