Answer:

There is no remainder because x - 2 is a factor of x⁴ - 3x³ + 4x² - 6x + 4

Step-by-step explanation:

Use either synthetic division or long division.

x⁴ - 3x³ + 4x² - 6x + 4 ÷ x - 2

Using Synthetic Division

Use the the divisor's root

x - 2 = 0 -> x = 2

Copy the coefficients in descending order

Bring down the 1st coefficient, multiply the root value to it then add to the 2nd coefficient.

Multiply the root value to it's sum then add to the next coefficient.

Continue until the last coefficient is used.

2| 1 -3 +4 -6 +4

| 2 -2 4 -4

1 -1 2 -2 0 -> last value (0)

remainder

To have the final answer, use the values of the bottom row as coefficients of the answer.

The final answer should be one less than the degree of the dividend. because of the polynomial degree 1 of the divisor.

The last value is the remainder. If the last value is 0, there is no remainder. The divisor is a factor of the dividend.

Result:

x³ - x² + 2x - 2