Divide as indicated.
$\left(16 x^3-5 x-17\right) \div(4 x-5)$
Answer (1)
Perform polynomial long division.
Divide 16 x 3 by 4 x to get 4 x 2 .
Continue the long division process to find the quotient and remainder.
The result is 4 x 2 + 5 x + 5 + 4 x − 5 8 .
Explanation
Understanding the Problem We are asked to divide the polynomial 16 x 3 − 5 x − 17 by the binomial 4 x − 5 . This is a polynomial long division problem. The expression ( 16 x 3 − 5 x − 17 ) ∗ ( 4 x − 5 ) is not part of the division problem, and seems to be an error. We will ignore it and focus on the division.
Objective We will perform polynomial long division of 16 x 3 − 5 x − 17 by 4 x − 5 .
First Term of Quotient First, we set up the long division problem. We write the dividend as 16 x 3 + 0 x 2 − 5 x − 17 to include the missing x 2 term. Then we divide the leading term of the dividend ( 16 x 3 ) by the leading term of the divisor ( 4 x ) to find the first term of the quotient. This is 4 x 16 x 3 = 4 x 2 .
Subtracting and Bringing Down Next, we multiply the divisor ( 4 x − 5 ) by the first term of the quotient ( 4 x 2 ) to get 4 x 2 ( 4 x − 5 ) = 16 x 3 − 20 x 2 . We subtract this from the dividend: ( 16 x 3 + 0 x 2 − 5 x − 17 ) − ( 16 x 3 − 20 x 2 ) = 20 x 2 − 5 x − 17 .
Second Term of Quotient Now we divide the leading term of the new dividend ( 20 x 2 ) by the leading term of the divisor ( 4 x ) to find the next term of the quotient. This is 4 x 20 x 2 = 5 x .
Subtracting Again We multiply the divisor ( 4 x − 5 ) by the second term of the quotient ( 5 x ) to get 5 x ( 4 x − 5 ) = 20 x 2 − 25 x . We subtract this from the new dividend: ( 20 x 2 − 5 x − 17 ) − ( 20 x 2 − 25 x ) = 20 x − 17 .
Third Term of Quotient We divide the leading term of the new dividend ( 20 x ) by the leading term of the divisor ( 4 x ) to find the next term of the quotient. This is 4 x 20 x = 5 .
Final Subtraction We multiply the divisor ( 4 x − 5 ) by the third term of the quotient ( 5 ) to get 5 ( 4 x − 5 ) = 20 x − 25 . We subtract this from the new dividend: ( 20 x − 17 ) − ( 20 x − 25 ) = 8 .
Final Result The remainder is 8 . Therefore, the result of the division is 4 x 2 + 5 x + 5 + 4 x − 5 8 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, in control systems, engineers use polynomial division to analyze the stability and performance of feedback systems. By dividing the characteristic polynomial of the system by a desired polynomial, they can determine the system's response and adjust its parameters to meet specific requirements. This ensures that the system operates efficiently and reliably.
