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. $x^2+x+1+\frac{2}{x-1}$
D. $x^3-x^2+x$
Answer (2)
Set up the synthetic division with the coefficients of the dividend (1, 0, 0, 1) and the root of the divisor (1).
Perform synthetic division to obtain the coefficients of the quotient and the remainder.
Write the quotient based on the result of the synthetic division: x 2 + x + 1 .
The quotient is x 2 + x + 1 + x − 1 2 .
Explanation
Setting up Synthetic Division We are asked to use synthetic division to find the quotient of ( x 3 + 1 ) ÷ ( x − 1 ) . First, we set up the synthetic division. The dividend is x 3 + 1 , which can be written as x 3 + 0 x 2 + 0 x + 1 . The coefficients are 1, 0, 0, and 1. The divisor is x − 1 , so we use 1 as the root.
Performing Synthetic Division Now, we perform 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.
Determining the Quotient and Remainder The quotient is 1 x 2 + 1 x + 1 , or x 2 + x + 1 . The remainder is 2, so we can write the result as x 2 + x + 1 + x − 1 2 .
Identifying the Quotient The quotient is x 2 + x + 1 .
Examples
Polynomial division, like the one we just performed, is useful in various engineering and scientific applications. For example, when designing a filter for signal processing, engineers often need to decompose a complex transfer function into simpler parts using polynomial division. This simplifies the analysis and implementation of the filter. Also, in control systems, polynomial division helps in determining the stability and response characteristics of a system.
Using synthetic division, we find that the quotient of ( x 3 + 1 ) ÷ ( x − 1 ) is x 2 + x + 1 + x − 1 2 . Therefore, the correct answer is option C. Synthetic division confirms that the coefficients lead to this result with a remainder of 2.
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