What is the quotient of [tex]$\left(3 x^4-4 x^2+8 x-1\right) \div(x-2)$[/tex]?
Answer (1)
Perform polynomial long division of ( 3 x 4 − 4 x 2 + 8 x − 1 ) by ( x − 2 ) .
The quotient is 3 x 3 + 6 x 2 + 8 x + 24 .
The remainder is 47 .
The result of the division is 3 x 3 + 6 x 2 + 8 x + 24 + x − 2 47 , so the quotient is 3 x 3 + 6 x 2 + 8 x + 24 .
Explanation
Understanding the Problem We are asked to find the quotient of the division problem ( 3 x 4 − 4 x 2 + 8 x − 1 ) ÷ ( x − 2 ) . This means we need to perform polynomial long division or synthetic division to find the quotient.
Setting up Long Division Let's perform polynomial long division. First, we write the dividend as 3 x 4 + 0 x 3 − 4 x 2 + 8 x − 1 to include the missing x 3 term. We are dividing by x − 2 .
First Step of Long Division Dividing 3 x 4 by x gives 3 x 3 . Multiply 3 x 3 by ( x − 2 ) to get 3 x 4 − 6 x 3 . Subtract this from the dividend: ( 3 x 4 + 0 x 3 − 4 x 2 + 8 x − 1 ) − ( 3 x 4 − 6 x 3 ) = 6 x 3 − 4 x 2 + 8 x − 1
Second Step of Long Division Now, divide 6 x 3 by x to get 6 x 2 . Multiply 6 x 2 by ( x − 2 ) to get 6 x 3 − 12 x 2 . Subtract this from the remaining polynomial: ( 6 x 3 − 4 x 2 + 8 x − 1 ) − ( 6 x 3 − 12 x 2 ) = 8 x 2 + 8 x − 1
Third Step of Long Division Next, divide 8 x 2 by x to get 8 x . Multiply 8 x by ( x − 2 ) to get 8 x 2 − 16 x . Subtract this from the remaining polynomial: ( 8 x 2 + 8 x − 1 ) − ( 8 x 2 − 16 x ) = 24 x − 1
Fourth Step of Long Division Finally, divide 24 x by x to get 24 . Multiply 24 by ( x − 2 ) to get 24 x − 48 . Subtract this from the remaining polynomial: ( 24 x − 1 ) − ( 24 x − 48 ) = 47
Final Result The quotient is 3 x 3 + 6 x 2 + 8 x + 24 and the remainder is 47 . Therefore, the result of the division is 3 x 3 + 6 x 2 + 8 x + 24 + x − 2 47 .
Identifying the Quotient The quotient of the division ( 3 x 4 − 4 x 2 + 8 x − 1 ) ÷ ( x − 2 ) is 3 x 3 + 6 x 2 + 8 x + 24 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and solving differential equations. For example, when designing a control system, engineers often use polynomial division to simplify transfer functions, making it easier to analyze and implement the system. Imagine you're designing a cruise control system for a car. You might use polynomial division to simplify the mathematical model of the car's engine and speed, allowing you to create a more efficient and stable control system. This simplification helps in predicting the car's response to different inputs and tuning the control parameters for optimal performance.
