Divide the following: [tex]$\frac{4 x^3+3 x^2+2 x-3}{x^2-2}$[/tex]

Since the denominator is a [$\square$] and nonlinear, we will use long division to solve this problem.

Perform polynomial long division.
Divide 4 x 3 by x 2 to get 4 x , multiply 4 x by x 2 − 2 and subtract from the dividend.
Divide 3 x 2 by x 2 to get 3 , multiply 3 by x 2 − 2 and subtract from the updated dividend.
Express the result as the quotient plus the remainder over the divisor: 4 x + 3 + x 2 − 2 10 x + 3 ​ ​

Explanation

Understanding the Problem We are asked to divide the polynomial 4 x 3 + 3 x 2 + 2 x − 3 by the polynomial x 2 − 2 . Since the denominator is a nonlinear polynomial, we will use polynomial long division to solve this problem.

Setting up Long Division We set up the long division as follows:


x^2 - 2 | 4x^3 + 3x^2 + 2x - 3


First Term of Quotient We divide the leading term of the dividend, 4 x 3 , by the leading term of the divisor, x 2 , to get 4 x . This is the first term of our quotient. We then multiply the divisor, x 2 − 2 , by 4 x to get 4 x 3 − 8 x . We subtract this from the dividend:

4x
x^2 - 2 | 4x^3 + 3x^2 + 2x - 3
-(4x^3 - 8x)
---------------------
3x^2 + 10x - 3


Second Term of Quotient Next, we divide the leading term of the new dividend, 3 x 2 , by the leading term of the divisor, x 2 , to get 3 . This is the next term of our quotient. We then multiply the divisor, x 2 − 2 , by 3 to get 3 x 2 − 6 . We subtract this from the new dividend:

4x + 3
x^2 - 2 | 4x^3 + 3x^2 + 2x - 3
-(4x^3 - 8x)
---------------------
3x^2 + 10x - 3
-(3x^2 - 6)
---------------------
10x + 3


Final Result The degree of the remainder, 10 x + 3 , is less than the degree of the divisor, x 2 − 2 , so we are done. The quotient is 4 x + 3 and the remainder is 10 x + 3 . We can express the result as:

4 x + 3 + x 2 − 2 10 x + 3 ​

Conclusion Therefore, the result of dividing 4 x 3 + 3 x 2 + 2 x − 3 by x 2 − 2 is 4 x + 3 + x 2 − 2 10 x + 3 ​ .

Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and data analysis. For example, in control systems, engineers use polynomial division to simplify transfer functions and analyze system stability. 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 speed response, making it easier to design a controller that keeps the car at the desired speed.

Answered by GinnyAnswer | 2025-07-08