The question requires us to find the values of $a$ and $b$ such that the polynomial $2x^3 - 5x^2 + ax + b$ has $(x - 2)$ and $x$ as its factors. Here's the step-by-step solution:

Step 1: Understanding the Problem

If $(x - 2)$ and $x$ are factors of the polynomial, then the polynomial must satisfy:

1. $P(2) = 0$
2. $P(0) = 0$

Here, $P(x) = 2x^3 - 5x^2 + ax + b$.

Step 2: Substitute $x = 0$

From $P(0) = 0$: $P(0) = 2(0)^3 - 5(0)^2 + a(0) + b = b$ This implies: $b = 0$

Step 3: Substitute $x = 2$

From $P(2) = 0$: $P(2) = 2(2)^3 - 5(2)^2 + a(2) + b = 0$ Substitute $b = 0$: $P(2) = 2(8) - 5(4) + 2a = 0$ Simplify: $16 - 20 + 2a = 0$ $-4 + 2a = 0$ $2a = 4$ $a = 2$

Final Answer

The values of $a$ and $b$ are: $a = 2, \quad b = 0$

Would you like further details or have any questions about the solution?
Here are some related questions to consider:

1. How do we determine the factors of a polynomial?
2. What is the significance of setting $P(x) = 0$ for the factors of a polynomial?
3. Can there be more than two factors for a cubic polynomial?
4. How would the solution change if $(x - 2)$ was not a factor?
5. How can synthetic division help verify $(x - 2)$ as a factor?

Tip: Always test each potential factor by substituting its root into the polynomial. This ensures the correctness of your calculations!