Use the remainder theorem to factorise the expression `2x^(3) + x^(2) + 7x-6`. Hence, solve the equation `2x^(3) + 9x^(2) + 7x-6=0`

Answer

To solve the problem using the Remainder Theorem and factorize the expression \(2x^3 + 9x^2 + 7x - 6\), we will follow these steps:

### Step 1: Define the polynomial
Let \( p(x) = 2x^3 + 9x^2 + 7x - 6 \).

### Step 2: Use the Remainder Theorem
We will check for possible rational roots using the Rational Root Theorem. We will test some integer values for \( x \).

### Step 3: Test \( x = -2 \)
Calculate \( p(-2) \):
\[ p(-2) = 2(-2)^3 + 9(-2)^2 + 7(-2) - 6 \]
\[ = 2(-8) + 9(4) - 14 - 6 \]
\[ = -16 + 36 - 14 - 6 \]
\[ = 0 \]
Since \( p(-2) = 0 \), \( x + 2 \) is a factor of \( p(x) \).

### Step 4: Perform polynomial long division
Now we will divide \( p(x) \) 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 \( p(x) \):
\[ (2x^3 + 9x^2 + 7x - 6) - (2x^3 + 4x^2) = 5x^2 + 7x - 6 \]

4. Now divide \( 5x^2 \) by \( x \):
\[ \frac{5x^2}{x} = 5x \]
5. Multiply \( 5x \) by \( x + 2 \):
\[ 5x(x + 2) = 5x^2 + 10x \]
6. Subtract:
\[ (5x^2 + 7x - 6) - (5x^2 + 10x) = -3x - 6 \]

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

So, we have:
\[ p(x) = (x + 2)(2x^2 + 5x - 3) \]

### Step 5: Factor \( 2x^2 + 5x - 3 \)
Now we need to factor \( 2x^2 + 5x - 3 \) using the middle term splitting method.

1. Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is -3):
\[ 2 \times -3 = -6 \]
2. We need two numbers that multiply to -6 and add to 5. The numbers are 6 and -1.
3. Rewrite the middle term:
\[ 2x^2 + 6x - x - 3 \]
4. Group the terms:
\[ (2x^2 + 6x) + (-x - 3) \]
5. Factor by grouping:
\[ 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) \]

### Step 6: Complete factorization
Thus, the complete factorization of \( p(x) \) is:
\[ p(x) = (x + 2)(2x - 1)(x + 3) \]

### Step 7: Solve the equation \( 2x^3 + 9x^2 + 7x - 6 = 0 \)
Set the factors to zero:
1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( 2x - 1 = 0 \) → \( x = \frac{1}{2} \)
3. \( x + 3 = 0 \) → \( x = -3 \)

### Final Solutions
The solutions to the equation \( 2x^3 + 9x^2 + 7x - 6 = 0 \) are:
\[ x = -2, \quad x = \frac{1}{2}, \quad x = -3 \]