Divide using long division.

$(4 x^3-2 x^2+2 x+2) \div(x+3)$

Perform polynomial long division of ( 4 x 3 − 2 x 2 + 2 x + 2 ) by ( x + 3 ) .
Divide 4 x 3 by x to get 4 x 2 , multiply ( x + 3 ) by 4 x 2 to get 4 x 3 + 12 x 2 , and subtract from the original polynomial to get − 14 x 2 + 2 x + 2 .
Divide − 14 x 2 by x to get − 14 x , multiply ( x + 3 ) by − 14 x to get − 14 x 2 − 42 x , and subtract from the remaining polynomial to get 44 x + 2 .
Divide 44 x by x to get 44 , multiply ( x + 3 ) by 44 to get 44 x + 132 , and subtract from the remaining polynomial to get a remainder of − 130 . The quotient is 4 x 2 − 14 x + 44 and the remainder is − 130 , so the answer is 4 x 2 − 14 x + 44 , R − 130 ​ .

Explanation

Understanding the Problem We want to divide the polynomial 4 x 3 − 2 x 2 + 2 x + 2 by x + 3 using long division. This process will give us a quotient and a remainder.

First Step of Long Division First, we set up the long division. We divide the leading term of the dividend, 4 x 3 , by the leading term of the divisor, x . This gives us 4 x 2 , which is the first term of the quotient.

Subtracting the First Term Next, we multiply the divisor, x + 3 , by 4 x 2 to get 4 x 3 + 12 x 2 . We subtract this from the dividend: ( 4 x 3 − 2 x 2 + 2 x + 2 ) − ( 4 x 3 + 12 x 2 ) = − 14 x 2 + 2 x + 2

Finding the Second Term of the Quotient Now, we divide the leading term of the new dividend, − 14 x 2 , by the leading term of the divisor, x . This gives us − 14 x , which is the next term of the quotient.

Subtracting the Second Term We multiply the divisor, x + 3 , by − 14 x to get − 14 x 2 − 42 x . We subtract this from the new dividend: ( − 14 x 2 + 2 x + 2 ) − ( − 14 x 2 − 42 x ) = 44 x + 2

Finding the Third Term of the Quotient Next, we divide the leading term of the new dividend, 44 x , by the leading term of the divisor, x . This gives us 44 , which is the next term of the quotient.

Subtracting the Third Term We multiply the divisor, x + 3 , by 44 to get 44 x + 132 . We subtract this from the new dividend: ( 44 x + 2 ) − ( 44 x + 132 ) = − 130

Final Result The remainder is − 130 . Therefore, the result of the division is 4 x 2 − 14 x + 44 with a remainder of − 130 . We can write this as: 4 x 2 − 14 x + 44 − x + 3 130 ​


Examples
Polynomial long division is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, when designing a digital filter, engineers use polynomial division to simplify the transfer function of the filter, making it easier to analyze and implement. Understanding polynomial division helps in optimizing system performance and ensuring stability.

Answered by GinnyAnswer | 2025-07-08