Divide using long division.
$(x^3-3 x^2-3 x-4) \div(x+2)$
Answer (1)
Perform polynomial long division of ( x 3 − 3 x 2 − 3 x − 4 ) by ( x + 2 ) .
Divide x 3 by x to get the first term of the quotient: x 2 .
Multiply ( x + 2 ) by x 2 and subtract from the dividend to get a new dividend: − 5 x 2 − 3 x − 4 .
Repeat the process to find the quotient x 2 − 5 x + 7 and the remainder − 18 .
The final answer is x 2 − 5 x + 7 with a remainder of − 18 , which can be written as x 2 − 5 x + 7 , R − 18 .
Explanation
Understanding the Problem We are asked to divide the polynomial x 3 − 3 x 2 − 3 x − 4 by the polynomial x + 2 using long division. Our goal is to find the quotient and the remainder of this division.
Setting up Long Division Let's set up the long division problem. We write the dividend ( x 3 − 3 x 2 − 3 x − 4 ) inside the division symbol and the divisor ( x + 2 ) outside.
First Term of Quotient Now, we divide the leading term of the dividend ( x 3 ) by the leading term of the divisor ( x ). This gives us x 2 , which is the first term of the quotient. x 3 ÷ x = x 2
Multiplying the Divisor Next, we multiply the divisor ( x + 2 ) by the first term of the quotient ( x 2 ). This gives us x 3 + 2 x 2 .
x 2 ( x + 2 ) = x 3 + 2 x 2
Subtracting We subtract this result from the dividend: ( x 3 − 3 x 2 − 3 x − 4 ) − ( x 3 + 2 x 2 ) = − 5 x 2 − 3 x − 4 .
Second Term of Quotient Now, we bring down the next term (-3x) from the dividend. We divide the leading term of the new dividend ( − 5 x 2 ) by the leading term of the divisor ( x ). This gives us − 5 x , which is the next term of the quotient. − 5 x 2 ÷ x = − 5 x
Multiplying the Divisor We multiply the divisor ( x + 2 ) by the new term of the quotient ( − 5 x ). This gives us − 5 x 2 − 10 x .
− 5 x ( x + 2 ) = − 5 x 2 − 10 x
Subtracting We subtract this result from the new dividend: ( − 5 x 2 − 3 x − 4 ) − ( − 5 x 2 − 10 x ) = 7 x − 4 .
Third Term of Quotient We bring down the next term (-4) from the dividend. We divide the leading term of the new dividend ( 7 x ) by the leading term of the divisor ( x ). This gives us 7 , which is the next term of the quotient. 7 x ÷ x = 7
Multiplying the Divisor We multiply the divisor ( x + 2 ) by the new term of the quotient ( 7 ). This gives us 7 x + 14 .
7 ( x + 2 ) = 7 x + 14
Subtracting We subtract this result from the new dividend: ( 7 x − 4 ) − ( 7 x + 14 ) = − 18 .
Final Result The quotient is x 2 − 5 x + 7 and the remainder is − 18 . Therefore, the result of the division is x 2 − 5 x + 7 + x + 2 − 18 .
Conclusion Thus, ( x 3 − 3 x 2 − 3 x − 4 ) ÷ ( x + 2 ) = x 2 − 5 x + 7 with a remainder of − 18 .
Examples
Polynomial long division is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, it can be used to analyze control systems or electrical circuits. Suppose you have a system described by a transfer function that is a ratio of two polynomials. Using polynomial long division, you can simplify the transfer function to analyze the system's behavior more easily. This simplification helps in designing controllers or predicting the system's response to different inputs. Understanding polynomial division allows engineers to model and optimize complex systems effectively.
