Use synthetic division to solve $(x^3+1) \div (x-1)$. What is the quotient?
A. $x^2+x+1+\frac{2}{x+1}$
B. $x^2-x+1$
C. $x^2+x+1+\frac{9}{x-1}$
D. $x^3-x^2+x$
Answer (2)
Set up the synthetic division with the coefficients of the dividend x 3 + 1 (1, 0, 0, 1) and the divisor x − 1 (using 1).
Perform the synthetic division to obtain the coefficients of the quotient and the remainder.
The coefficients of the quotient are 1, 1, and 1, so the quotient is x 2 + x + 1 .
The remainder is 2, so the complete result is x 2 + x + 1 + x − 1 2 . The quotient is x 2 + x + 1 .
Explanation
Understanding the problem We are asked to divide the polynomial x 3 + 1 by the polynomial x − 1 using synthetic division and find the quotient.
Setting up synthetic division To set up the synthetic division, we write the coefficients of the dividend x 3 + 1 , which are 1, 0, 0, and 1. The divisor is x − 1 , so we use 1 as the divisor in the synthetic division.
Performing synthetic division Performing the synthetic division:
1 | 1 0 0 1
| 1 1 1
------------
1 1 1 2
The numbers 1, 1, and 1 are the coefficients of the quotient, and 2 is the remainder.
Finding the quotient and remainder The quotient is 1 x 2 + 1 x + 1 , which is x 2 + x + 1 . The remainder is 2. Therefore, the result of the division is x 2 + x + 1 + x − 1 2 .
Identifying the quotient The quotient is 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 the characteristic equation and determine the system's poles, which indicate whether the system is stable or unstable. This helps in designing controllers that ensure the system operates reliably and efficiently.
Using synthetic division on ( x 3 + 1 ) ÷ ( x − 1 ) gives the quotient x 2 + x + 1 . The result is obtained by analyzing the coefficients and calculating the remainder. The final answer is option B.
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