Write the expression for a cubic function f if f(1)=6 and f(-1)=f(0)=f(2)=0

To find the cubic function \( f(x) \) that satisfies the conditions \( f(1) = 6 \) and \( f(-1) = f(0) = f(2) = 0 \), we can follow these steps:

### Step 1: General Form of a Cubic Function
A general cubic function can be written as:
\[
f(x) = ax^3 + bx^2 + cx + d
\]

### Step 2: Apply the Given Conditions
Since \( f(-1) = 0 \), \( f(0) = 0 \), and \( f(2) = 0 \), the function has roots at \( x = -1 \), \( x = 0 \), and \( x = 2 \). Therefore, \( f(x) \) can be expressed as:
\[
f(x) = a(x + 1)(x)(x - 2)
\]
Here, \( a \) is a constant coefficient that we need to determine.

### Step 3: Use the Remaining Condition
We know that \( f(1) = 6 \). Substitute \( x = 1 \) into the equation to find \( a \):
\[
f(1) = a(1 + 1)(1)(1 - 2) = 6
\]
Simplifying this:
\[
f(1) = a(2)(1)(-1) = -2a
\]
\[
-2a = 6
\]
\[
a = -3
\]

### Step 4: Write the Final Expression
Substitute \( a = -3 \) back into the function:
\[
f(x) = -3(x + 1)(x)(x - 2)
\]
Expanding this expression:
\[
f(x) = -3(x^3 - x^2 - 2x)
\]
So the function is:
\[
f(x) = -3x^3 + 3x^2 + 6x
\]

Thus, the cubic function is:
\[
f(x) = -3x^3 + 3x^2 + 6x
\]

Would you like more details on any part of this process or have any other questions?

Here are 5 related questions you might consider:
1. How do you find the roots of a cubic equation?
2. What is the significance of the constant \( a \) in the cubic function?
3. Can this function be factored further?
4. How do cubic functions differ from quadratic functions in terms of their graphs?
5. What are some real-world applications of cubic functions?

**Tip:** When solving for coefficients in polynomial functions, always check by substituting values back into the original equation to verify accuracy.

Discover how to find the cubic function f(x) satisfying f(1)=6 and additional conditions such as f(-1)=f(0)=f(2)=0. Learn about polynomial functions and solving cubic equations with step-by-step guidance. Suitable for advanced high school students.

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