Answer: (x^4 - 1) ÷ (x - 1) is x^3 - x^2 + x - 1
Explanation:
To solve the expression (x^4 - 1) ÷ (x - 1) using synthetic division, you need to follow these steps:
1. Write down the coefficients of the dividend (x^4 - 1) in descending order: 1, 0, 0, 0, -1.
2. Set up the synthetic division table by writing the divisor (x - 1) to the left of the table and placing the first coefficient of the dividend (1) in the top row of the table.
3. Divide the first coefficient (1) by the divisor (x - 1). The result is the first term of the quotient, which is x^3.
4. Multiply the divisor (x - 1) by the first term of the quotient (x^3) and write the product below the second coefficient of the dividend (0).
5. Subtract the product from the second coefficient (0) of the dividend and write the result below it.
6. Repeat steps 3-5 with the new number below the line until you reach the last coefficient of the dividend (-1).
7. The numbers in the bottom row of the table represent the coefficients of the quotient, from highest degree to lowest. In this case, the quotient is x^3 - x^2 + x - 1.