The problem states:

• $(x - 2)$ is a factor of $2x^3 + ax^2 + bx - 2$.
• When the expression is divided by $(x + 3)$, the remainder is $-50$.
• You are required to find the values of $a$ and $b$.

Let me solve this step by step.

Step 1: Using the fact that $x - 2$ is a factor

Since $x - 2$ is a factor, substituting $x = 2$ into the polynomial $P(x)$ gives: $P(2) = 0.$

The polynomial is $P(x) = 2x^3 + ax^2 + bx - 2$. Substituting $x = 2$: $2(2)^3 + a(2)^2 + b(2) - 2 = 0.$ $16 + 4a + 2b - 2 = 0.$ 4a + 2b + 14 = 0. \tag{1}

Step 2: Using the remainder when divided by $(x + 3)$

The remainder when the polynomial is divided by $(x + 3)$ is $-50$. Substituting $x = -3$ into $P(x)$, we get: $P(-3) = -50.$

Substituting $x = -3$ into $P(x) = 2x^3 + ax^2 + bx - 2$: $2(-3)^3 + a(-3)^2 + b(-3) - 2 = -50.$ $2(-27) + a(9) + b(-3) - 2 = -50.$ $-54 + 9a - 3b - 2 = -50.$ $9a - 3b - 56 = -50.$ 9a - 3b - 6 = 0. \tag{2}

Step 3: Solving the system of equations

From Equation (1): 4a + 2b + 14 = 0 \quad \Rightarrow \quad 2a + b = -7. \tag{3}

From Equation (2): 9a - 3b - 6 = 0 \quad \Rightarrow \quad 3a - b = 2. \tag{4}

Adding Equations (3) and (4): $2a + b + 3a - b = -7 + 2.$ $5a = -5 \quad \Rightarrow \quad a = -1.$

Substitute $a = -1$ into Equation (3): $2(-1) + b = -7.$ $-2 + b = -7 \quad \Rightarrow \quad b = -5.$

Final Answer

The values of $a$ and $b$ are: $\boxed{a = -1 \text{ and } b = -5.}$

Would you like a more detailed breakdown of the steps, or do you have further questions?

Related Questions

1. How do you confirm if a polynomial is divisible by a factor like $(x - c)$?
2. What is the Remainder Theorem, and how is it applied in this problem?
3. Can you graphically represent this polynomial with $a = -1$ and $b = -5$?
4. How do changes in coefficients $a$ and $b$ affect the behavior of the polynomial?
5. What other methods can be used to solve polynomial factorization problems?

Tip

Always substitute given values into polynomials carefully to avoid sign errors, especially when working with cube terms!