https://file.aiquickdraw.com/mathbot/file/1724997460028ra88ydxj.jpg

given f(x)​=x³+cx²+c, find f(c)+f(-c)

To solve the problem in the image, we need to calculate \( f(c) + f(-c) \) where \( f(x) = x^3 + cx^2 + c \).

### Step 1: Calculate \( f(c) \)
Substitute \( x = c \) into the function:
\[
f(c) = c^3 + c(c^2) + c = c^3 + c^3 + c = 2c^3 + c
\]

### Step 2: Calculate \( f(-c) \)
Substitute \( x = -c \) into the function:
\[
f(-c) = (-c)^3 + c(-c)^2 + c = -c^3 + c(c^2) + c = -c^3 + c^3 + c = c
\]

### Step 3: Find \( f(c) + f(-c) \)
Add the results from Step 1 and Step 2:
\[
f(c) + f(-c) = (2c^3 + c) + c = 2c^3 + 2c
\]

### Conclusion
The correct answer is \( \boxed{2c^3 + 2c} \).

Would you like more details or have any questions about this solution? Here are some related questions:

1. What happens if the constant \( c \) equals 0 in the function?
2. How would the solution change if the function were \( f(x) = x^3 + cx + c \)?
3. Can you factor the expression \( 2c^3 + 2c \)?
4. What is the significance of symmetry in the function values \( f(c) \) and \( f(-c) \)?
5. How would you compute \( f(2c) + f(-2c) \)?

**Tip:** When substituting negative values into polynomials, pay careful attention to the signs to avoid errors in calculation.

Learn how to calculate f(c) + f(-c) for the cubic function f(x) = x^3 + cx^2 + c with a step-by-step solution. This problem explores polynomial evaluation and symmetry in function values. Suitable for advanced high school students.

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