The problem asks us to find the values of $a$ and $b$ in the expression $ax^3 - x^2 + bx - 1$, given that when divided by $x+2$ the remainder is $-33$, and when divided by $x-3$ the remainder is $77$. We are then asked to find the remainder when the expression is divided by $x-2$.

1. Problem Description

The problem asks us to find the values of $a$ and $b$ in the expression $ax^3 - x^2 + bx - 1$, given that when divided by $x+2$ the remainder is $-33$, and when divided by $x-3$ the remainder is $77$. We are then asked to find the remainder when the expression is divided by $x-2$.

2. Solution Steps

We use the Remainder Theorem, which states that if we divide a polynomial $f(x)$ by $x-c$, the remainder is $f(c)$.

* Step 1: Apply the Remainder Theorem for $x+2$

Since dividing by $x+2$ gives a remainder of $-33$, we have $f(-2) = -33$. Substituting $x = -2$ into the expression gives:

$a(-2)^3 - (-2)^2 + b(-2) - 1 = -33$

$-8a - 4 - 2b - 1 = -33$

$-8a - 2b - 5 = -33$

$-8a - 2b = -28$

$4a + b = 14$ (Equation 1)

* Step 2: Apply the Remainder Theorem for $x-3$

Since dividing by $x-3$ gives a remainder of $77$, we have $f(3) = 77$. Substituting $x = 3$ into the expression gives:

$a(3)^3 - (3)^2 + b(3) - 1 = 77$

$27a - 9 + 3b - 1 = 77$

$27a + 3b - 10 = 77$

$27a + 3b = 87$

$9a + b = 29$ (Equation 2)

* Step 3: Solve the system of equations

We have the following system of linear equations:

$4a + b = 14$

$9a + b = 29$

Subtracting Equation 1 from Equation 2, we get:

$(9a + b) - (4a + b) = 29 - 14$

$5a = 15$

$a = 3$

Substituting $a=3$ into Equation 1:

$4(3) + b = 14$

$12 + b = 14$

$b = 2$

* Step 4: Find the remainder when divided by $x-2$

Now that we know $a=3$ and $b=2$, the expression is $3x^3 - x^2 + 2x - 1$.

We want to find the remainder when divided by $x-2$.

Using the Remainder Theorem, we need to find $f(2)$:

$f(2) = 3(2)^3 - (2)^2 + 2(2) - 1$

$f(2) = 3(8) - 4 + 4 - 1$

$f(2) = 24 - 4 + 4 - 1$

$f(2) = 23$

3. Final Answer

$a = 3$, $b = 2$, and the remainder when divided by $x-2$ is

2

3.