Use the Remainder Theorem to factorise the following expression `2x^(3)+x^(2)-13x+6`.

Answer

To factorize the expression \(2x^3 + x^2 - 13x + 6\) using the Remainder Theorem, follow these steps:

### Step 1: Define the polynomial
Let \( f(x) = 2x^3 + x^2 - 13x + 6 \).

### Step 2: Use the Remainder Theorem
To find a root of the polynomial, we can test some rational values. Let's start by testing \( x = 2 \).

### Step 3: Evaluate \( f(2) \)
Calculate \( f(2) \):
\[ f(2) = 2(2)^3 + (2)^2 - 13(2) + 6 \]
Calculating each term:
- \( 2(2)^3 = 2 \times 8 = 16 \)
- \( (2)^2 = 4 \)
- \( -13(2) = -26 \)
- The constant term is \( +6 \)

Now combine these:
\[ f(2) = 16 + 4 - 26 + 6 = 0 \]
Since \( f(2) = 0 \), \( x - 2 \) is a factor of the polynomial.

### Step 4: Perform polynomial long division
Now, divide \( f(x) \) by \( x - 2 \).

1. Divide the leading term: \( \frac{2x^3}{x} = 2x^2 \).
2. Multiply \( 2x^2 \) by \( x - 2 \):
\[ 2x^2(x - 2) = 2x^3 - 4x^2 \]
3. Subtract this from \( f(x) \):
\[ (2x^3 + x^2 - 13x + 6) - (2x^3 - 4x^2) = 5x^2 - 13x + 6 \]

4. Now, divide the leading term of the new polynomial: \( \frac{5x^2}{x} = 5x \).
5. Multiply \( 5x \) by \( x - 2 \):
\[ 5x(x - 2) = 5x^2 - 10x \]
6. Subtract:
\[ (5x^2 - 13x + 6) - (5x^2 - 10x) = -3x + 6 \]

7. Divide the leading term: \( \frac{-3x}{x} = -3 \).
8. Multiply \( -3 \) by \( x - 2 \):
\[ -3(x - 2) = -3x + 6 \]
9. Subtract:
\[ (-3x + 6) - (-3x + 6) = 0 \]

### Step 5: Write the factorization
Thus, we can express \( f(x) \) as:
\[ f(x) = (x - 2)(2x^2 + 5x - 3) \]

### Step 6: Factor the quadratic \( 2x^2 + 5x - 3 \)
To factor \( 2x^2 + 5x - 3 \), we can split the middle term:
\[ 2x^2 - x + 6x - 3 \]
Grouping:
\[ = x(2x - 1) + 3(2x - 1) = (2x - 1)(x + 3) \]

### Final Factorization
Thus, the complete factorization of the polynomial is:
\[ f(x) = (x - 2)(2x - 1)(x + 3) \]