The given problem states:

"If $(2x - 1)$ is a factor of the polynomial $P(x) = 2x^3 - 5x^2 - kx + 3$, find the value of $k$. Find the other two factors of $P(x)$."

Step 1: Use the Factor Theorem

Since $(2x - 1)$ is a factor of the polynomial $P(x)$, according to the factor theorem, the root of $2x - 1 = 0$, which is $x = \frac{1}{2}$, should satisfy $P\left( \frac{1}{2} \right) = 0$.

Let's substitute $x = \frac{1}{2}$ into the polynomial $P(x)$:

$P(x) = 2x^3 - 5x^2 - kx + 3$

Substitute $x = \frac{1}{2}$:

$P\left( \frac{1}{2} \right) = 2\left( \frac{1}{2} \right)^3 - 5\left( \frac{1}{2} \right)^2 - k\left( \frac{1}{2} \right) + 3$

Simplify each term:

$P\left( \frac{1}{2} \right) = 2\left( \frac{1}{8} \right) - 5\left( \frac{1}{4} \right) - \frac{k}{2} + 3$ $P\left( \frac{1}{2} \right) = \frac{2}{8} - \frac{5}{4} - \frac{k}{2} + 3$ $P\left( \frac{1}{2} \right) = \frac{1}{4} - \frac{5}{4} - \frac{k}{2} + 3$

Now combine like terms:

$P\left( \frac{1}{2} \right) = -\frac{4}{4} - \frac{k}{2} + 3$ $P\left( \frac{1}{2} \right) = -1 - \frac{k}{2} + 3$ $P\left( \frac{1}{2} \right) = 2 - \frac{k}{2}$

Since $P\left( \frac{1}{2} \right) = 0$, we set the expression equal to zero:

$2 - \frac{k}{2} = 0$

Step 2: Solve for $k$

$2 = \frac{k}{2}$ Multiply both sides by 2: $k = 4$

Thus, the value of $k$ is $4$.

Step 3: Find the other two factors

Now that we know $k = 4$, the polynomial becomes:

$P(x) = 2x^3 - 5x^2 - 4x + 3$

We are given that $(2x - 1)$ is one factor. To find the other factors, we can perform polynomial division of $P(x)$ by $2x - 1$.

Perform the division:

1. Divide the leading term $2x^3$ by $2x$ to get $x^2$.
2. Multiply $x^2$ by $2x - 1$, which gives $2x^3 - x^2$.
3. Subtract this from the original polynomial:

$(2x^3 - 5x^2 - 4x + 3) - (2x^3 - x^2) = -4x^2 - 4x + 3$

4. Divide $-4x^2$ by $2x$ to get $-2x$.
5. Multiply $-2x$ by $2x - 1$, which gives $-4x^2 + 2x$.
6. Subtract:

$(-4x^2 - 4x + 3) - (-4x^2 + 2x) = -6x + 3$

7. Divide $-6x$ by $2x$ to get $-3$.
8. Multiply $-3$ by $2x - 1$, which gives $-6x + 3$.
9. Subtract:

$(-6x + 3) - (-6x + 3) = 0$

Thus, the quotient from the division is $x^2 - 2x - 3$.

Step 4: Factor the quotient

Now we factor $x^2 - 2x - 3$. This factors as:

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

Final Answer:

The complete factorization of $P(x)$ is:

$P(x) = (2x - 1)(x - 3)(x + 1)$

Would you like further details or clarifications? Here are some related questions:

1. How is the factor theorem applied to solve polynomial factorization problems?
2. Can all cubic polynomials be factored into linear factors?
3. How does polynomial division help in finding factors of higher-degree polynomials?
4. What other methods can be used to factor polynomials besides division and the factor theorem?
5. How can you check if your factorization of a polynomial is correct?

Tip: When factoring a cubic polynomial, always try the factor theorem first if you know one factor—it simplifies the process of finding the remaining factors.