What are the quotient and remainder when $5 x^4-2 x^2$ is divided by $x^3-x^2+1$?

The quotient is $\square$ and the remainder is $\square$.

When dividing 5 x 4 − 2 x 2 by x 3 − x 2 + 1 , the quotient is 5 x + 5 and the remainder is 3 x 2 − 5 x − 5 .
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Answered by Anonymous | 2025-07-04

Perform polynomial long division of 5 x 4 − 2 x 2 by x 3 − x 2 + 1 .
Identify the quotient from the long division, which is 5 x + 5 .
Identify the remainder from the long division, which is 3 x 2 − 5 x − 5 .
State the quotient and remainder: 5 x + 5 ​ and 3 x 2 − 5 x − 5 ​ .

Explanation

Problem Analysis We are asked to find the quotient and remainder when 5 x 4 − 2 x 2 is divided by x 3 − x 2 + 1 . This is a polynomial long division problem.

Polynomial Long Division We perform polynomial long division as follows:


5x + 5
x^3-x^2+1 | 5x^4 + 0x^3 - 2x^2 + 0x + 0
- (5x^4 - 5x^3 + 0x^2 + 5x)
----------------------------
5x^3 - 2x^2 - 5x + 0
- (5x^3 - 5x^2 + 0x + 5)
----------------------------
3x^2 - 5x - 5

Therefore, when 5 x 4 − 2 x 2 is divided by x 3 − x 2 + 1 , the quotient is 5 x + 5 and the remainder is 3 x 2 − 5 x − 5 .

Final Answer Thus, the quotient is 5 x + 5 and the remainder is 3 x 2 − 5 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, polynomial division can help simplify transfer functions, making it easier to analyze system stability and design controllers. In signal processing, it can be used to decompose signals into simpler components. Understanding polynomial division is also crucial in coding theory for error detection and correction.

Answered by GinnyAnswer | 2025-07-04