Given the polynomial function: $$f(x) = ax^3 - 11x^2 + bx + 4$$, where $a$ and $b$ are constants - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 5

Given that $(3x + 2)$ is a factor of $f(x)$, we can perform polynomial long division to factorize the function completely:

1. Dividing the polynomial:

   Let:

   $f(x) = (3x + 2)(Ax^2 + Bx + C)$

   We need to find coefficients $A$, $B$, and $C$. Substituting, we have:

   = 3Ax^3 + (2A + 3B)x^2 + (2B + 3C)x + 2C$$ Comparing coefficients of $f(x)$ with the expanded form: - For $x^3$: $3A = a = 6$. Hence, $A = 2.$ - For $x^2$: $2A + 3B = -11

ightarrow 4 + 3B = -11 ightarrow 3B = -15 ightarrow B = -5.$

• For $x^1$: $2B + 3C = -4 ightarrow 2(-5) + 3C = -4 ightarrow -10 + 3C = -4 ightarrow 3C = 6 ightarrow C = 2.$

2. Final factorization:

   Thus, we can express $f(x)$ as:

   $f(x) = (3x + 2)(2x^2 - 5x + 2)$

3. Factoring the quadratic:

   The quadratic can be factored as:

   $2x^2 - 5x + 2 = (2x - 1)(x - 2).$

4. Complete factorization of $f(x)$:

   Thus:

   $f(x) = (3x + 2)(2x - 1)(x - 2).$