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To perform synthetic division for the expression x⁷-125÷x-2, we need to divide the polynomial by x-2.

First, let's set up the synthetic division table:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

Next, we bring down the coefficient of the highest power of x, which is 1:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

Next, we multiply the divisor, 2, by the result we just obtained (1), and write the product below the next coefficient:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2

Next, we add the next coefficient (0) to the product we just obtained (2):

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2 0

We repeat this process for each coefficient:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2 0 0

4 0 0 0

8 0 0 0 0

16 0 0 0 0 0

32 0 0 0 0 0 0

64 0 0 0 0 0 0 0

The final row represents the coefficients of the quotient polynomial. Therefore, the quotient is x⁶ + 2x⁵ + 4x⁴ + 8x³ + 16x² + 32x + 64.

Note: The remainder is 0, indicating that x-2 is a factor of x⁷-125.

First, let's set up the synthetic division table:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

Next, we bring down the coefficient of the highest power of x, which is 1:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

Next, we multiply the divisor, 2, by the result we just obtained (1), and write the product below the next coefficient:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2

Next, we add the next coefficient (0) to the product we just obtained (2):

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2 0

We repeat this process for each coefficient:

2 | 1 0 0 0 0 0 0 -125

|_____________________________

1

2 0 0

4 0 0 0

8 0 0 0 0

16 0 0 0 0 0

32 0 0 0 0 0 0

64 0 0 0 0 0 0 0

The final row represents the coefficients of the quotient polynomial. Therefore, the quotient is x⁶ + 2x⁵ + 4x⁴ + 8x³ + 16x² + 32x + 64.

Note: The remainder is 0, indicating that x-2 is a factor of x⁷-125.

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