We are given the expression $ax^3 - x^2 + bx - 1$. When this expression is divided by $x+2$, the remainder is -33. When it is divided by $x-3$, the remainder is 77. We need to find the values of $a$ and $b$, and then find the remainder when the expression is divided by $x-2$.

1. Problem Description

We are given the expression $ax^3 - x^2 + bx - 1$. When this expression is divided by $x+2$, the remainder is -

3

3. When it is divided by $x-3$, the remainder is

7

7. We need to find the values of $a$ and $b$, and then find the remainder when the expression is divided by $x-2$.

2. Solution Steps

According to the Remainder Theorem, if a polynomial $f(x)$ is divided by $x-c$, the remainder is $f(c)$.

When $ax^3 - x^2 + bx - 1$ is divided by $x+2$, the remainder is -

3

3. Therefore,

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

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

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

$-8a - 2b = -28$

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

When $ax^3 - x^2 + bx - 1$ is divided by $x-3$, the remainder is

7

7. Therefore,

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

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

$27a + 3b - 10 = 77$

$27a + 3b = 87$

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

Subtract Equation 1 from Equation 2:

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

$5a = 15$

$a = 3$

Substitute $a=3$ into Equation 1:

$4(3) + b = 14$

$12 + b = 14$

$b = 2$

So, $a=3$ and $b=2$. The expression is $3x^3 - x^2 + 2x - 1$.

Now, we need to find the remainder when $3x^3 - x^2 + 2x - 1$ is divided by $x-2$.

Let $x=2$.

Remainder = $3(2)^3 - (2)^2 + 2(2) - 1 = 3(8) - 4 + 4 - 1 = 24 - 4 + 4 - 1 = 23$.

3. Final Answer

$a = 3$, $b = 2$, and the remainder is

2

3.