We are given the expression $ax^3 - x^2 + bx - 1$. When this expression is divided by $x+2$ and $x-3$, the remainders are -33 and 77 respectively. 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 $x+2$ and $x-3$, the remainders are -33 and 77 respectively. 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 $f(c)$.

Step 1: Use the remainder when divided by $x+2$.

Let $f(x) = ax^3 - x^2 + bx - 1$.

When $f(x)$ is divided by $x+2$, the remainder is -

3

3. Thus, $f(-2) = -33$.

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

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

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

$-8a - 2b = -28$

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

Step 2: Use the remainder when divided by $x-3$.

When $f(x)$ is divided by $x-3$, the remainder is

7

7. Thus, $f(3) = 77$.

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

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

$27a + 3b - 10 = 77$

$27a + 3b = 87$

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

Step 3: Solve for $a$ and $b$.

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 $b=2$.

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

The expression is now $3x^3 - x^2 + 2x - 1$.

When divided by $x-2$, the remainder is $f(2)$.

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

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

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

$f(2) = 23$

3. Final Answer

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

2

3.