Use synthetic division to solve $(x^3-x^2-17 x-15)
\div (x-5)$. What is the quotient?

Using synthetic division, we find that when dividing x 3 − x 2 − 17 x − 15 by x − 5 , the quotient is x 2 + 4 x + 3 . The process involves setting up coefficients, performing multiplication and addition with the divisor's root, and interpreting the results. Ultimately, the resulting quotient of the division is x 2 + 4 x + 3 .
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Answered by Anonymous | 2025-07-04

Set up the synthetic division table with the coefficients of the polynomial and the root of the divisor.
Perform synthetic division to find the coefficients of the quotient and the remainder.
Write the quotient using the coefficients obtained from the synthetic division.
The quotient is x 2 + 4 x + 3 ​ .

Explanation

Problem Analysis We are given the polynomial x 3 − x 2 − 17 x − 15 and asked to divide it by ( x − 5 ) using synthetic division. Our goal is to find the quotient of this division.

Setting up Synthetic Division We set up the synthetic division with the coefficients of the polynomial x 3 − x 2 − 17 x − 15 , which are 1 , − 1 , − 17 , and − 15 . The root of the divisor ( x − 5 ) is 5 .

Performing Synthetic Division Now, we perform synthetic division:

Bring down the first coefficient: 1 .

Multiply the root 5 by 1 and add it to the second coefficient − 1 : 5 × 1 + ( − 1 ) = 4 .

Multiply the root 5 by 4 and add it to the third coefficient − 17 : 5 × 4 + ( − 17 ) = 3 .

Multiply the root 5 by 3 and add it to the last coefficient − 15 : 5 × 3 + ( − 15 ) = 0 .


The resulting numbers are 1 , 4 , 3 , and 0 . The last number, 0 , is the remainder. The other numbers are the coefficients of the quotient.

Determining the Quotient The coefficients of the quotient are 1 , 4 , and 3 . Since we divided a cubic polynomial by a linear term, the quotient is a quadratic polynomial. Therefore, the quotient is 1 x 2 + 4 x + 3 , which simplifies to x 2 + 4 x + 3 .

Final Answer Thus, the quotient of the division x − 5 x 3 − x 2 − 17 x − 15 ​ is x 2 + 4 x + 3 .


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 instance, 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 system's behavior. This helps in designing stable and efficient control systems for various applications, from robotics to aerospace.

Answered by GinnyAnswer | 2025-07-04