Question 1 The polynomial $P(x)=x^{4}-2 x^{3}+a x+b$ has remainder 3 after division by $(x-1)$, and has remainder -5 after division by $(x+1)$. Find $a$ and $b$.

Step 1 ：Given the polynomial \(P(x) = x^4 - 2x^3 + ax + b\), we know that it has a remainder of 3 when divided by \((x-1)\) and a remainder of -5 when divided by \((x+1)\).

Step 2 ：Using the Remainder Theorem, we can find the values of a and b by evaluating the polynomial at x = 1 and x = -1:

Step 3 ：\(P(1) = 1^4 - 2(1)^3 + a(1) + b = 3\)

Step 4 ：\(P(-1) = (-1)^4 - 2(-1)^3 + a(-1) + b = -5\)

Step 5 ：Solving the system of equations, we get:

Step 6 ：\(a + b - 1 = 3\)

Step 7 ：\(-a + b + 3 = -5\)

Step 8 ：Solving for a and b, we find that \(a = 6\) and \(b = -2\).

Step 9 ：\(\boxed{a = 6, b = -2}\)