What are the quotient and remainder when $5 x^{4}-3 x^{2}$ is divided by $x^{3}-x^{2}+2 ?$
The quotient is and the remainder is
Answer
\(\boxed{\text{Final Answer: The quotient is } x \text{ and the remainder is } 5x^{2} - 3x^{2} + 2x}\)
Step 1 ：We are asked to find the quotient and remainder when the polynomial \(5x^{4}-3x^{2}\) is divided by the polynomial \(x^{3}-x^{2}+2\).
Step 2 ：We can solve this problem using polynomial division.
Step 3 ：Let's denote the first polynomial as \(f\) and the second polynomial as \(g\). So, \(f = 5x^{4} - 3x^{2}\) and \(g = x^{3} - x^{2} + 2\).
Step 4 ：Performing the division, we find that the quotient is \(x\) and the remainder is \(5x^{2} - 3x^{2} + 2x\).
Step 5 ：\(\boxed{\text{Final Answer: The quotient is } x \text{ and the remainder is } 5x^{2} - 3x^{2} + 2x}\)
