Use synthetic division to solve $(x^4-1) \div(x-1)$. What is the quotient?
A. $x^3-x^2+x-1$
B. $x^3$
C. $x^3+x^2+x+1$
D. $x^3-2$
Answer (1)
Set up the synthetic division table with the coefficients of the dividend x 4 − 1 and the divisor x − 1 .
Perform synthetic division.
Identify the coefficients of the quotient from the result of the synthetic division.
Write the quotient using the coefficients obtained: x 3 + x 2 + x + 1 .
Explanation
Understanding the Problem We are asked to divide the polynomial x 4 − 1 by x − 1 using synthetic division and determine the quotient.
Setting up Synthetic Division To perform synthetic division, we set up a table using the coefficients of the dividend x 4 − 1 , which are 1, 0, 0, 0, -1. The divisor is x − 1 , so we use 1 as the divisor in the synthetic division.
Performing Synthetic Division We perform synthetic division as follows:
Write down the coefficients of the dividend: 1 0 0 0 -1
Write the divisor (1) to the left.
Bring down the first coefficient (1).
Multiply the divisor (1) by the number brought down (1) to get 1, and write it under the second coefficient (0).
Add the second coefficient (0) and the result from the previous step (1) to get 1.
Multiply the divisor (1) by the result from the previous step (1) to get 1, and write it under the third coefficient (0).
Add the third coefficient (0) and the result from the previous step (1) to get 1.
Multiply the divisor (1) by the result from the previous step (1) to get 1, and write it under the fourth coefficient (0).
Add the fourth coefficient (0) and the result from the previous step (1) to get 1.
Multiply the divisor (1) by the result from the previous step (1) to get 1, and write it under the fifth coefficient (-1).
Add the fifth coefficient (-1) and the result from the previous step (1) to get 0.
The last row is 1 1 1 1 0.
Determining the Quotient The numbers in the last row, excluding the last number, are the coefficients of the quotient. The coefficients are 1, 1, 1, 1. Therefore, the quotient is 1 x 3 + 1 x 2 + 1 x + 1 , which simplifies to x 3 + x 2 + x + 1 .
Final Answer The quotient of the polynomial division ( x 4 − 1 ) ÷ ( x − 1 ) is x 3 + x 2 + x + 1 .
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . It's often used in engineering to simplify complex polynomial expressions that arise in control systems, signal processing, and structural analysis. For example, when analyzing the stability of a control system, engineers might use synthetic division to factor out known roots of a characteristic equation, making it easier to determine the remaining roots and assess system stability. This technique streamlines calculations and provides quick insights into system behavior.
