If (x-2) is a factor of `2x^(3)-x^(2)-px-2`
(i) find the value of p.
(ii) with the value of p, factorise the above expression completely.

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Identify the polynomial
Let the polynomial be:
\[ f(x) = 2x^3 - x^2 - px - 2 \]

### Step 2: Use the Factor Theorem
Since \( (x - 2) \) is a factor of \( f(x) \), we know that:
\[ f(2) = 0 \]

### Step 3: Substitute \( x = 2 \) into the polynomial
Now, we will substitute \( x = 2 \) into the polynomial:
\[ f(2) = 2(2)^3 - (2)^2 - p(2) - 2 \]
Calculating each term:
\[ = 2(8) - 4 - 2p - 2 \]
\[ = 16 - 4 - 2p - 2 \]
\[ = 10 - 2p \]

### Step 4: Set the equation to zero
Since \( f(2) = 0 \):
\[ 10 - 2p = 0 \]

### Step 5: Solve for \( p \)
Rearranging the equation gives:
\[ 2p = 10 \]
\[ p = 5 \]

### Step 6: Substitute \( p \) back into the polynomial
Now that we have \( p = 5 \), we substitute it back into the polynomial:
\[ f(x) = 2x^3 - x^2 - 5x - 2 \]

### Step 7: Factor the polynomial
Now we will factor \( f(x) = 2x^3 - x^2 - 5x - 2 \) using synthetic division or polynomial long division by \( (x - 2) \).

1. Divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).
2. Multiply \( (x - 2) \) by \( 2x^2 \) to get \( 2x^3 - 4x^2 \).
3. Subtract this from the original polynomial:
\[ (-x^2 + 4x^2) = 3x^2 \]
4. Bring down the next term:
\[ 3x^2 - 5x \]
5. Divide \( 3x^2 \) by \( x \) to get \( 3x \).
6. Multiply \( (x - 2) \) by \( 3x \) to get \( 3x^2 - 6x \).
7. Subtract:
\[ (-5x + 6x) = x \]
8. Bring down the last term:
\[ x - 2 \]
9. Divide \( x \) by \( x \) to get \( 1 \).
10. Multiply \( (x - 2) \) by \( 1 \) to get \( x - 2 \).
11. Subtract:
\[ 0 \]

### Step 8: Write the complete factorization
Thus, we have:
\[ f(x) = (x - 2)(2x^2 + 3x + 1) \]

### Step 9: Factor \( 2x^2 + 3x + 1 \)
Now we will factor \( 2x^2 + 3x + 1 \):
1. The factors of \( 2 \) and \( 1 \) that add up to \( 3 \) are \( 2 \) and \( 1 \).
2. Rewrite:
\[ 2x^2 + 2x + x + 1 \]
3. Group:
\[ 2x(x + 1) + 1(x + 1) \]
4. Factor out \( (x + 1) \):
\[ (x + 1)(2x + 1) \]

### Final Factorization
Thus, the complete factorization of the polynomial is:
\[ f(x) = (x - 2)(x + 1)(2x + 1) \]

### Summary of Steps
1. Identify the polynomial and use the Factor Theorem.
2. Substitute \( x = 2 \) to find \( p \).
3. Substitute \( p \) back into the polynomial.
4. Use polynomial long division to factor the polynomial.
5. Factor the resulting quadratic completely.