The problem states:

"When $ax^3 - x^2 + 2x + b$ is divided by $x - 1$, the remainder is 10. When it is divided by $x - 2$, the remainder is 51. Find $a$ and $b$."

Step 1: Using the Remainder Theorem

According to the Remainder Theorem, if a polynomial $f(x)$ is divided by $x - c$, the remainder is $f(c)$. We will use this for both conditions provided in the problem.

Let: $f(x) = ax^3 - x^2 + 2x + b$

Condition 1: When $f(x)$ is divided by $x - 1$

The remainder is 10, so: $f(1) = a(1)^3 - (1)^2 + 2(1) + b = 10$ Simplifying: $a - 1 + 2 + b = 10$ $a + b + 1 = 10$ $a + b = 9 \quad \text{(Equation 1)}$

Condition 2: When $f(x)$ is divided by $x - 2$

The remainder is 51, so: $f(2) = a(2)^3 - (2)^2 + 2(2) + b = 51$ Simplifying: $8a - 4 + 4 + b = 51$ $8a + b = 51 \quad \text{(Equation 2)}$

Step 2: Solving the System of Equations

We have the following system of linear equations:

1. $a + b = 9$
2. $8a + b = 51$

Subtract Equation 1 from Equation 2: $(8a + b) - (a + b) = 51 - 9$ $7a = 42$ $a = 6$

Now, substitute $a = 6$ into Equation 1: $6 + b = 9$ $b = 3$

Solution

The values of $a$ and $b$ are: $a = 6$ $b = 3$

Would you like more details or have any questions?

Here are some questions that might expand on this topic:

1. How does the Remainder Theorem help in solving polynomial equations?
2. What would happen if the values of the remainder were different?
3. Can you explain why this method works for any linear divisor of the form $x - c$?
4. How can these techniques be applied to higher-degree polynomials?
5. What other methods could be used to find the values of $a$ and $b$ in this problem?

Tip: Always double-check your values by substituting them back into the original conditions to ensure they satisfy all the requirements.