Select the correct answer.
Using long division, what is the quotient of $3 x^4+20 x^3+14 x^2+17 x+30$ and $x+6 ?$
A. $3 x^2+2 x+2$
B. $3 x^3+2 x^2+2 x+5$
C. $2 x^3+14 x^2+17 x+30$
D. $3 x^3-2 x^2+2 x-5$
Answer (1)
Perform polynomial long division of 3 x 4 + 20 x 3 + 14 x 2 + 17 x + 30 by x + 6 .
Divide 3 x 4 by x to get 3 x 3 , multiply ( x + 6 ) by 3 x 3 to get 3 x 4 + 18 x 3 , and subtract from the original polynomial.
Continue the long division process to find the quotient.
The quotient is 3 x 3 + 2 x 2 + 2 x + 5 , so the answer is 3 x 3 + 2 x 2 + 2 x + 5 .
Explanation
Problem Analysis We are asked to find the quotient of the polynomial division ( 3 x 4 + 20 x 3 + 14 x 2 + 17 x + 30 ) / ( x + 6 ) using long division.
First Step of Long Division We perform polynomial long division. First, we divide 3 x 4 by x to get 3 x 3 . Then we multiply ( x + 6 ) by 3 x 3 to get 3 x 4 + 18 x 3 . Subtracting this from the original polynomial gives us: ( 3 x 4 + 20 x 3 + 14 x 2 + 17 x + 30 ) − ( 3 x 4 + 18 x 3 ) = 2 x 3 + 14 x 2 + 17 x + 30
Second Step of Long Division Next, we divide 2 x 3 by x to get 2 x 2 . Then we multiply ( x + 6 ) by 2 x 2 to get 2 x 3 + 12 x 2 . Subtracting this from the remainder gives us: ( 2 x 3 + 14 x 2 + 17 x + 30 ) − ( 2 x 3 + 12 x 2 ) = 2 x 2 + 17 x + 30
Third Step of Long Division Next, we divide 2 x 2 by x to get 2 x . Then we multiply ( x + 6 ) by 2 x to get 2 x 2 + 12 x . Subtracting this from the remainder gives us: ( 2 x 2 + 17 x + 30 ) − ( 2 x 2 + 12 x ) = 5 x + 30
Fourth Step of Long Division Finally, we divide 5 x by x to get 5 . Then we multiply ( x + 6 ) by 5 to get 5 x + 30 . Subtracting this from the remainder gives us: ( 5 x + 30 ) − ( 5 x + 30 ) = 0
Final Result The quotient is therefore 3 x 3 + 2 x 2 + 2 x + 5 .
Verification The coefficients of the quotient are [3, 2, 2, 5], which corresponds to the polynomial 3 x 3 + 2 x 2 + 2 x + 5 .
Examples
Polynomial long 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 control systems. By dividing the characteristic polynomial of the system by a desired polynomial, they can determine the system's response and stability margins. This helps them design controllers that ensure the system operates reliably and meets performance requirements.
