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We are given the polynomial function \( f(x) = 2x^3 + x^2 - 18x - 9 \) and the information that \( (x + 3) \) is a factor. We are required to factorize the polynomial.

Since \( (x + 3) \) is a factor, we can perform polynomial division to divide \( f(x) \) by \( (x + 3) \). Let's proceed with the division.

### Step 1: Polynomial Division
We'll divide \( 2x^3 + x^2 - 18x - 9 \) by \( x + 3 \).

1. **Divide the first term**: \( \frac{2x^3}{x} = 2x^2 \).
2. **Multiply**: \( 2x^2 \cdot (x + 3) = 2x^3 + 6x^2 \).
3. **Subtract**: \((2x^3 + x^2) - (2x^3 + 6x^2) = -5x^2\).
4. **Bring down the next term**: \( -5x^2 - 18x \).
5. **Divide the first term**: \( \frac{-5x^2}{x} = -5x \).
6. **Multiply**: \( -5x \cdot (x + 3) = -5x^2 - 15x \).
7. **Subtract**: \((-5x^2 - 18x) - (-5x^2 - 15x) = -3x \).
8. **Bring down the next term**: \( -3x - 9 \).
9. **Divide the first term**: \( \frac{-3x}{x} = -3 \).
10. **Multiply**: \( -3 \cdot (x + 3) = -3x - 9 \).
11. **Subtract**: \((-3x - 9) - (-3x - 9) = 0\).

The quotient is \( 2x^2 - 5x - 3 \), and the remainder is 0, confirming that \( (x + 3) \) is indeed a factor.

### Step 2: Factorizing the Quotient
Now, we need to factor the quadratic \( 2x^2 - 5x - 3 \).

To factorize \( 2x^2 - 5x - 3 \), we can use the middle-term splitting method.

1. Find two numbers whose product is \( 2 \times -3 = -6 \) and whose sum is \( -5 \).
   - The numbers are \( -6 \) and \( 1 \) because \( (-6) + 1 = -5 \) and \( (-6) \times 1 = -6 \).
2. Rewrite the middle term:
   \[
   2x^2 - 6x + x - 3
   \]
3. Factor by grouping:
   \[
   2x(x - 3) + 1(x - 3)
   \]
4. Factor out the common factor:
   \[
   (2x + 1)(x - 3)
   \]

### Final Factorization
The full factorization of \( f(x) = 2x^3 + x^2 - 18x - 9 \) is:
\[
f(x) = (x + 3)(2x + 1)(x - 3)
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you determine the roots of the polynomial \( f(x) = 2x^3 + x^2 - 18x - 9 \) after factorization?
2. How can synthetic division be used to divide a polynomial instead of polynomial division?
3. What is the significance of the remainder being zero in polynomial division?
4. How can the factor theorem be applied to check if a binomial is a factor of a polynomial?
5. What other methods can be used to factorize cubic polynomials?

**Tip:** When factorizing a polynomial, always check if simple substitutions or factor theorem can be used before jumping into complex methods.

Given that (x + 3) is a factor, factorise f(x) = 2x^3 + x^2 - 18x - 9.

Learn how to factorize the cubic polynomial f(x) = 2x^3 + x^2 - 18x - 9 given that (x + 3) is a factor. This problem covers polynomial division, the factor theorem, and the factorization of quadratic equations. Suitable for students in Grades 10-12.

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