Use synthetic division to find the quotient and remainder.
$\frac{3 x^3+2 x-5}{x-1}$
quotient $\square$
remainder $\square$
Answer (2)
Using synthetic division on the polynomial 3 x 3 + 2 x − 5 divided by x − 1 , we find that the quotient is 3 x 2 + 3 x + 5 and the remainder is 0. The process involves setting up coefficients and employing multiplication and addition at each step. This method simplifies polynomial division, especially for higher degree polynomials.
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Set up the synthetic division with the coefficients of the polynomial and the root of the divisor.
Perform the synthetic division process.
Identify the coefficients of the quotient from the result of the synthetic division.
Determine the remainder from the last number in the result.
The quotient is 3 x 2 + 3 x + 5 and the remainder is 0 .
Explanation
Understanding the Problem We are given the polynomial 3 x 3 + 2 x − 5 and we want to divide it by x − 1 using synthetic division. Our goal is to find the quotient and the remainder of this division.
Setting up Synthetic Division First, we set up the synthetic division table. We write down the coefficients of the polynomial 3 x 3 + 2 x − 5 , which are 3 , 0 (since there is no x 2 term), 2 , and − 5 . We also write down the root of the divisor x − 1 , which is 1 .
Performing Synthetic Division Now, we perform synthetic division:
Bring down the first coefficient, which is 3 .
Multiply the root 1 by 3 to get 3 , and write it below the second coefficient 0 .
Add 0 and 3 to get 3 .
Multiply the root 1 by 3 to get 3 , and write it below the third coefficient 2 .
Add 2 and 3 to get 5 .
Multiply the root 1 by 5 to get 5 , and write it below the last coefficient − 5 .
Add − 5 and 5 to get 0 .
Identifying Quotient and Remainder The numbers in the last row, excluding the last number, are the coefficients of the quotient. In this case, the coefficients are 3 , 3 , and 5 . Since we started with a cubic polynomial and divided by a linear term, the quotient will be a quadratic polynomial. Thus, the quotient is 3 x 2 + 3 x + 5 . The last number in the last row is the remainder, which is 0 .
Final Answer Therefore, when we divide 3 x 3 + 2 x − 5 by x − 1 , the quotient is 3 x 2 + 3 x + 5 and the remainder is 0 .
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 polynomial, making it easier to determine the remaining roots and assess system behavior. This helps in designing stable and efficient control systems.
