Answer:Therefore, the expression can be written as q(x) = 3x^2 - x + 1 + 123/(x - 1).
Step-by-step explanation:To use synthetic division, we will divide the polynomial 3x^3 + 2x^2 + 113x^3 + 2x^2 + 11 by the polynomial x + 1.
1. Write the coefficients of the polynomial in descending order: 3, 2, 113, 2, 11.
2. Write the divisor in the form (x + a), where a is the opposite of the constant term: x + (-1) = x - 1.
3. Set up the synthetic division table, with the coefficients of the polynomial as the first row and the divisor as the second row:
-1 | 3 2 113 2 11
4. Bring down the first coefficient (3) and multiply it by the divisor's constant term (-1). Write the result (3) in the next row:
-1 | 3 2 113 2 11
-3
5. Add the two numbers in the second row to get the next term (2 + (-3) = -1). Write it in the next row:
-1 | 3 2 113 2 11
-3 -1
6. Multiply the constant term of the divisor (-1) by the result in the second row (-1). Write the result (1) in the next row:
-1 | 3 2 113 2 11
-3 -1 1
7. Add the two numbers in the third row to get the next term (113 + 1 = 114). Write it in the next row:
-1 | 3 2 113 2 11
-3 -1 1 114
8. Repeat the process with the next term (2):
-1 | 3 2 113 2 11
-3 -1 1 114 112
9. Finally, add the two numbers in the last row to get the remainder (11 + 112 = 123):
-1 | 3 2 113 2 11
-3 -1 1 114 112
123
The result of the division is:
3x^2 - x + 1 with a remainder of 123.
Therefore, the expression can be written as q(x) = 3x^2 - x + 1 + 123/(x - 1).To use synthetic division, we will divide the polynomial 3x^3 + 2x^2 + 113x^3 + 2x^2 + 11 by the polynomial x + 1.
1. Write the coefficients of the polynomial in descending order: 3, 2, 113, 2, 11.
2. Write the divisor in the form (x + a), where a is the opposite of the constant term: x + (-1) = x - 1.
3. Set up the synthetic division table, with the coefficients of the polynomial as the first row and the divisor as the second row:
-1 | 3 2 113 2 11
4. Bring down the first coefficient (3) and multiply it by the divisor's constant term (-1). Write the result (3) in the next row:
-1 | 3 2 113 2 11
-3
5. Add the two numbers in the second row to get the next term (2 + (-3) = -1). Write it in the next row:
-1 | 3 2 113 2 11
-3 -1
6. Multiply the constant term of the divisor (-1) by the result in the second row (-1). Write the result (1) in the next row:
-1 | 3 2 113 2 11
-3 -1 1
7. Add the two numbers in the third row to get the next term (113 + 1 = 114). Write it in the next row:
-1 | 3 2 113 2 11
-3 -1 1 114
8. Repeat the process with the next term (2):
-1 | 3 2 113 2 11
-3 -1 1 114 112
9. Finally, add the two numbers in the last row to get the remainder (11 + 112 = 123):
-1 | 3 2 113 2 11
-3 -1 1 114 112
123
The result of the division is:
3x^2 - x + 1 with a remainder of 123.
Therefore, the expression can be written as q(x) = 3x^2 - x + 1 + 123/(x - 1).