Find an expression for a cubic function f if f(1) = 12 and
f(−5) = f(0) = f(2) = 0.

To find the expression for the cubic function \( f(x) \) given the conditions:

- \( f(1) = 12 \)
- \( f(-5) = 0 \)
- \( f(0) = 0 \)
- \( f(2) = 0 \)

### Step 1: General Form of the Cubic Function
A cubic function can generally be written as:
\[
f(x) = a(x - r_1)(x - r_2)(x - r_3)
\]
where \( r_1 \), \( r_2 \), and \( r_3 \) are the roots of the function, and \( a \) is a constant.

### Step 2: Using the Given Roots
We are given that \( f(-5) = 0 \), \( f(0) = 0 \), and \( f(2) = 0 \). Therefore, the roots of the function are \( x = -5 \), \( x = 0 \), and \( x = 2 \).

Thus, the function can be written as:
\[
f(x) = a(x + 5)x(x - 2)
\]

### Step 3: Use the Condition \( f(1) = 12 \)
Substitute \( x = 1 \) and \( f(1) = 12 \) into the function to find the constant \( a \).

\[
f(1) = a(1 + 5)(1)(1 - 2) = 12
\]

Simplifying:
\[
f(1) = a(6)(1)(-1) = -6a
\]

Set this equal to 12:
\[
-6a = 12
\]

Solve for \( a \):
\[
a = -2
\]

### Step 4: Write the Final Expression
Substitute \( a = -2 \) back into the function:
\[
f(x) = -2(x + 5)x(x - 2)
\]

### Final Answer
The expression for the cubic function is:
\[
f(x) = -2(x + 5)x(x - 2)
\]

Would you like more details or have any questions?

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Here are some related questions:
1. How do you find the roots of a cubic function?
2. What is the significance of the coefficient \( a \) in a cubic function?
3. Can a cubic function have more than three roots?
4. How do you determine if a cubic function has a local maximum or minimum?
5. What are the applications of cubic functions in real-world problems?

**Tip:** When finding the roots of a polynomial function, look for values of \( x \) that make the function equal to zero. These values are crucial in determining the structure of the function.

Learn how to find the expression for a cubic function f(x) given conditions such as f(1) = 12, f(-5) = 0, f(0) = 0, and f(2) = 0. This detailed solution covers the use of roots in cubic functions and Vieta's formulas, suitable for advanced high school students.

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