The curve C has equation $y = f(x)$, where $f'(x) = (x - 3)(3x + 5)$ Given that the point P(1, 20) lies on C, (a) find $f(x)$, simplifying each term - Edexcel - A-Level Maths Pure - Question 10 - 2018 - Paper 1

To express $f(x)$ in the form $f(x) = (x - 3)(x + A)$, we first factor out $(x - 3)$ from:

$f(x) = x^3 - 2x^2 - 15x + 36$

Using polynomial long division or synthetic division, we divide by $x - 3$:

When dividing, we find:

$A$ must equal 4, confirmed through coefficients comparison:

Thus, we can write:

$f(x) = (x - 3)(x^2 + Ax + B)$

After solving, we find:

$A = 4$

This shows:

$f(x) = (x - 3)(x + 4)$

To sketch the graph of C, we begin by identifying key points:

1. Intersections with the x-axis: Set $f(x) = 0$:

   $(x - 3)(x + 4) = 0$

   The solutions are:

   • $x = 3$
   • $x = -4$

2. Intersection with the y-axis: This occurs where $x = 0$:

   $f(0) = 36$

Thus, the coordinates for the intersections are:

• The curve cuts the x-axis at $(3, 0)$ and $(-4, 0)$.
• The curve cuts the y-axis at $(0, 36)$.

3. Sketch: Outline the curve keeping in mind it is a cubic function with local extrema and crossing through these points.