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To determine which function matches the given polynomial function, we'll compare the coefficients of each term. The polynomial we are comparing against is:

\[ f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \]

Let's compare this with each option:

a. \( f(x) = -10x^4 + 12x^3 - 12x^2 + 2x - 7 \)

b. \( f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \)

c. \( f(x) = -5x^4 + 12x^3 - 4x^2 + 2x - 7 \)

d. \( f(x) = 5x^4 + 12x^3 - 12x^2 + 2x - 7 \)

e. \( f(x) = 10x^4 + 12x^3 - 12x^2 + 2x - 7 \)

Now, let's analyze each option:

1. **Option a:**
   \[ f(x) = -10x^4 + 12x^3 - 12x^2 + 2x - 7 \]
   The coefficient of \(x^4\) is -10, which does not match -5.

2. **Option b:**
   \[ f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \]
   All coefficients match perfectly: -5, 12, -12, 2, -7.

3. **Option c:**
   \[ f(x) = -5x^4 + 12x^3 - 4x^2 + 2x - 7 \]
   The coefficient of \(x^2\) is -4, which does not match -12.

4. **Option d:**
   \[ f(x) = 5x^4 + 12x^3 - 12x^2 + 2x - 7 \]
   The coefficient of \(x^4\) is 5, which does not match -5.

5. **Option e:**
   \[ f(x) = 10x^4 + 12x^3 - 12x^2 + 2x - 7 \]
   The coefficient of \(x^4\) is 10, which does not match -5.

Therefore, the correct option is:

**b. \( f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \)**

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How do you factor a polynomial?
2. What is the derivative of the polynomial \( f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \)?
3. How do you find the roots of the polynomial \( f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 \)?
4. What is the significance of the leading coefficient in a polynomial?
5. How does the degree of a polynomial affect its graph?

**Tip:** When comparing polynomials, always ensure that the coefficients of corresponding terms match exactly.

Which function matches the polynomial f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7?

Learn how to identify which polynomial function matches the given function f(x) = -5x^4 + 12x^3 - 12x^2 + 2x - 7 by comparing the coefficients of each term. Suitable for students in Grades 9-12.

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