https://file.aiquickdraw.com/mathbot/file/1725379761719zvj3o1bf.jpg

Let's go through each part of the problem step by step.

### Part (a)
**Given that \((x - 1)\) is a factor of \(f(x)\), show that \(a + b + c = -1\).**

Since \((x - 1)\) is a factor of \(f(x)\), then \(f(1) = 0\). Substituting \(x = 1\) into the function:

\[
f(1) = 1^3 + a \cdot 1^2 + b \cdot 1 + c = 1 + a + b + c = 0.
\]

Therefore,

\[
a + b + c = -1.
\]

### Part (b)
**Find the values of \(a\), \(b\), and \(c\).**

From the problem, we are given the following conditions:

1. When \(f(x)\) is divided by \((x - 2)\), the remainder is \(-4\). This implies \(f(2) = -4\).
   
   Substituting \(x = 2\):

   \[
   f(2) = 2^3 + a \cdot 2^2 + b \cdot 2 + c = 8 + 4a + 2b + c = -4.
   \]

   Simplifying, we get:

   \[
   4a + 2b + c = -12. \quad \text{(Equation 1)}
   \]

2. When \(f(x)\) is divided by \((x - 3)\), the remainder is \(-12\). This implies \(f(3) = -12\).
   
   Substituting \(x = 3\):

   \[
   f(3) = 3^3 + a \cdot 3^2 + b \cdot 3 + c = 27 + 9a + 3b + c = -12.
   \]

   Simplifying, we get:

   \[
   9a + 3b + c = -39. \quad \text{(Equation 2)}
   \]

Now, we have three equations:

1. \(a + b + c = -1\) (from part a).
2. \(4a + 2b + c = -12\) (Equation 1).
3. \(9a + 3b + c = -39\) (Equation 2).

Let's solve these equations:

Subtract equation 1 from equation 2:

\[
(9a + 3b + c) - (4a + 2b + c) = -39 - (-12),
\]

which simplifies to:

\[
5a + b = -27. \quad \text{(Equation 3)}
\]

Subtract equation 1 from equation 3:

\[
(4a + 2b + c) - (a + b + c) = -12 - (-1),
\]

which simplifies to:

\[
3a + b = -11. \quad \text{(Equation 4)}
\]

Now, subtract equation 4 from equation 3:

\[
(5a + b) - (3a + b) = -27 - (-11),
\]

which simplifies to:

\[
2a = -16,
\]

so:

\[
a = -8.
\]

Substitute \(a = -8\) into equation 4:

\[
3(-8) + b = -11,
\]

which simplifies to:

\[
-24 + b = -11,
\]

so:

\[
b = 13.
\]

Substitute \(a = -8\) and \(b = 13\) into the original equation \(a + b + c = -1\):

\[
-8 + 13 + c = -1,
\]

which simplifies to:

\[
5 + c = -1,
\]

so:

\[
c = -6.
\]

Thus, \(a = -8\), \(b = 13\), and \(c = -6\).

### Part (c)
**Express \(f(x)\) as the product of three linear factors.**

We know \((x - 1)\) is a factor. Using the values \(a = -8\), \(b = 13\), and \(c = -6\), the cubic function is:

\[
f(x) = x^3 - 8x^2 + 13x - 6.
\]

To factor \(f(x)\), we divide by \((x - 1)\):

\[
f(x) = (x - 1)(x^2 - 7x + 6).
\]

Factoring the quadratic:

\[
x^2 - 7x + 6 = (x - 6)(x - 1).
\]

Thus:

\[
f(x) = (x - 1)^2 (x - 6).
\]

### Part (d)
**Sketch the graph of \(f(x)\).**

To sketch \(f(x)\):

- **Roots**: \(x = 1\) (with multiplicity 2) and \(x = 6\).
- **Y-intercept**: Substitute \(x = 0\):

  \[
  f(0) = -6.
  \]

Plot these points and sketch the curve with the shape expected from a cubic with a double root at \(x = 1\) (tangent to the x-axis) and crossing at \(x = 6\).

---

Would you like more details on any part? Here are some further explorations:

1. What happens if \((x - 2)\) was also a factor of \(f(x)\)?
2. How would the graph change if the coefficients were different?
3. Can you verify the factors of \(f(x)\) using synthetic division?
4. How can the nature of roots be verified algebraically?
5. What if the remainder theorem is used on a quadratic function?

**Tip:** Always check for simple factors first, such as \(x - 1\), when given roots or factors.

A cubic function is defined in terms of the constants a, b, and c as f(x) = x^3 + ax^2 + bx + c, where x is a real number. The following tasks are given:
(a) Given that (x - 1) is a factor of f(x), show that a + b + c = -1.
(b) It is further given that when f(x) is divided by (x - 2) the remainder is -4 and when f(x) is divided by (x - 3) the remainder is -12. Find the values of a, b, and c.
(c) Hence express f(x) as the product of three linear factors.
(d) Sketch the graph of f(x). The sketch must include the coordinates of any points where the graph of f(x) meets the coordinate axes.

This problem involves analyzing a cubic function f(x) = x^3 + ax^2 + bx + c, where you're tasked with finding the values of constants a, b, and c. The problem utilizes the Factor Theorem, Remainder Theorem, and involves factorizing the function and sketching its graph. Suitable for students in Grades 11-12.

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