*f*(*x*)=−8*x*3(*x*−6)3(*x*−4)2(*x*+8)3(*x*−2)3(*x*−3)4

Without graphing the function, answer each of the following questions.

a) What is the degree of ​​​​𝑓*f*​?  

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b) What is the leading coefficient of 𝑓*f*?​

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c) What is the maximum number of 𝑥*x*​-intercepts that the graph of 𝑓*f* can have?​

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d) What is the maximum number of turning points that the graph 𝑓*f* can have?​

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e) What is the 𝑦*y*​-intercept of 𝑓*f*​?  Write your answer as an ordered pair.

Let's analyze the function \( f(x) = -8x^3(x - 6)^3(x - 4)^2(x + 8)^3(x - 2)^3(x - 3)^4 \) to answer each question:

### a) What is the degree of \( f(x) \)?

The degree of a polynomial is the sum of the exponents of \( x \) in all the factors:

- \( x^3 \) contributes a degree of 3.
- \( (x - 6)^3 \) contributes a degree of 3.
- \( (x - 4)^2 \) contributes a degree of 2.
- \( (x + 8)^3 \) contributes a degree of 3.
- \( (x - 2)^3 \) contributes a degree of 3.
- \( (x - 3)^4 \) contributes a degree of 4.

Adding these together, the total degree of the polynomial is:
\[ 3 + 3 + 2 + 3 + 3 + 4 = 18 \]

**Answer**: The degree of \( f(x) \) is **18**.

### b) What is the leading coefficient of \( f(x) \)?

The leading coefficient is the coefficient of the term with the highest degree when the polynomial is expanded.

- The highest degree term is obtained by multiplying the leading terms of each factor:
\[ -8x^3 \cdot x^3 \cdot x^2 \cdot x^3 \cdot x^3 \cdot x^4 = -8x^{18} \]

The coefficient of this term is \(-8\).

**Answer**: The leading coefficient of \( f(x) \) is **-8**.

### c) What is the maximum number of \( x \)-intercepts that the graph of \( f(x) \) can have?

The \( x \)-intercepts occur where \( f(x) = 0 \). Each factor of the polynomial corresponds to a potential \( x \)-intercept:

- \( x^3 \) contributes one intercept at \( x = 0 \) (with multiplicity 3).
- \( (x - 6)^3 \) contributes one intercept at \( x = 6 \) (with multiplicity 3).
- \( (x - 4)^2 \) contributes one intercept at \( x = 4 \) (with multiplicity 2).
- \( (x + 8)^3 \) contributes one intercept at \( x = -8 \) (with multiplicity 3).
- \( (x - 2)^3 \) contributes one intercept at \( x = 2 \) (with multiplicity 3).
- \( (x - 3)^4 \) contributes one intercept at \( x = 3 \) (with multiplicity 4).

While these factors suggest 6 distinct intercepts, the multiplicities mean some intercepts might touch but not cross the x-axis.

**Answer**: The maximum number of \( x \)-intercepts the graph can have is **6**.

### d) What is the maximum number of turning points that the graph \( f(x) \) can have?

The maximum number of turning points of a polynomial function is one less than the degree of the polynomial.

Since the degree is 18:

**Answer**: The maximum number of turning points is \( 18 - 1 = 17 \).

### e) What is the \( y \)-intercept of \( f(x) \)? Write your answer as an ordered pair.

The \( y \)-intercept occurs where \( x = 0 \). To find the \( y \)-intercept, substitute \( x = 0 \) into the function:

\[
f(0) = -8(0)^3(0 - 6)^3(0 - 4)^2(0 + 8)^3(0 - 2)^3(0 - 3)^4
\]
Simplify each term:
\[
f(0) = -8(0)(-6)^3(-4)^2(8)^3(-2)^3(-3)^4
\]
Since \( (0)^3 = 0 \), the entire expression equals 0.

**Answer**: The \( y \)-intercept is \( (0, 0) \).

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Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How does the multiplicity of a root affect the shape of the graph near that root?
2. Why does a negative leading coefficient affect the end behavior of the polynomial?
3. How can we determine the end behavior of the polynomial based on its degree and leading coefficient?
4. What impact does the degree of a polynomial have on the number of \( x \)-intercepts?
5. What is the significance of the \( y \)-intercept when analyzing the graph of a function?

**Tip**: The degree of a polynomial tells you the maximum number of roots and turning points, but the exact number of each depends on the behavior of the function at those points!

Explore the polynomial f(x) = -8x^3(x - 6)^3(x - 4)^2(x + 8)^3(x - 2)^3(x - 3)^4 and answer questions about its degree, leading coefficient, x-intercepts, turning points, and y-intercept. Learn the concepts of polynomial degrees, leading coefficients, and graph behavior. Suitable for advanced high school students.

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