Using the factor theorem, show that (x-2) is a factor of `3x^(2)-5x-2`. Hence factorise the given expression.

Answer

To show that \( (x-2) \) is a factor of \( 3x^2 - 5x - 2 \) using the Factor Theorem, we will follow these steps:

### Step 1: Define the polynomial
Let \( p(x) = 3x^2 - 5x - 2 \).

### Step 2: Apply the Factor Theorem
According to the Factor Theorem, \( (x - c) \) is a factor of \( p(x) \) if \( p(c) = 0 \). Here, we will check if \( (x - 2) \) is a factor by substituting \( x = 2 \) into \( p(x) \).

### Step 3: Substitute \( x = 2 \) into \( p(x) \)
Calculate \( p(2) \):
\[ p(2) = 3(2)^2 - 5(2) - 2 \]
\[ = 3(4) - 10 - 2 \]
\[ = 12 - 10 - 2 \]
\[ = 12 - 12 \]
\[ = 0 \]

### Step 4: Conclusion from the Factor Theorem
Since \( p(2) = 0 \), we conclude that \( (x - 2) \) is indeed a factor of \( 3x^2 - 5x - 2 \).

### Step 5: Factor the polynomial
Now, we will factor \( 3x^2 - 5x - 2 \) using polynomial long division by dividing \( p(x) \) by \( (x - 2) \).

1. Divide the leading term: \( 3x^2 \div x = 3x \).
2. Multiply \( 3x \) by \( (x - 2) \):
\[ 3x(x - 2) = 3x^2 - 6x \]
3. Subtract this from \( p(x) \):
\[ (3x^2 - 5x - 2) - (3x^2 - 6x) = (-5x + 6x) - 2 = x - 2 \]
4. Now, divide \( x - 2 \) by \( (x - 2) \):
\[ (x - 2) \div (x - 2) = 1 \]
5. Therefore, the complete factorization is:
\[ 3x^2 - 5x - 2 = (x - 2)(3x + 1) \]

### Final Answer
Thus, the factorization of \( 3x^2 - 5x - 2 \) is:
\[ 3x^2 - 5x - 2 = (x - 2)(3x + 1) \]

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