PMGiven the polynomial function:a) Determine the number of turning points.b) Describe the end behavior.c) Use Descartes' Rule of Signs to determine the number of postive and negative real zeros.d) Find all the real zeros of the polynomial function. State any multiplicity.e) Sketch the graph.f(x)=x4-2x3-8x+16

Given the polynomial function:

\[
f(x) = x^4 - 2x^3 - 8x + 16
\]

We will address each part of the problem step by step:

### a) Determine the number of turning points
The number of turning points in a polynomial function is at most \( n - 1 \), where \( n \) is the degree of the polynomial. 

Since the degree of the polynomial \( f(x) \) is 4, the maximum number of turning points is:

\[
4 - 1 = 3
\]

Thus, the polynomial can have up to 3 turning points.

### b) Describe the end behavior
For end behavior, we focus on the leading term of the polynomial. The leading term is \( x^4 \), which is a positive even-degree term.

- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

In other words, both ends of the graph will rise towards positive infinity.

### c) Use Descartes' Rule of Signs to determine the number of positive and negative real zeros

#### For positive real zeros:
- We count the sign changes in \( f(x) = x^4 - 2x^3 - 8x + 16 \).
  - From \( x^4 \) to \( -2x^3 \), there is a sign change (positive to negative).
  - From \( -2x^3 \) to \( -8x \), no sign change.
  - From \( -8x \) to \( +16 \), there is a sign change (negative to positive).

Thus, there are **2 sign changes**, indicating that there are either 2 or 0 positive real zeros.

#### For negative real zeros:
- To apply Descartes' Rule of Signs for negative real zeros, substitute \( x \) with \( -x \) and simplify \( f(-x) \):
  
\[
f(-x) = (-x)^4 - 2(-x)^3 - 8(-x) + 16 = x^4 + 2x^3 + 8x + 16
\]

Now, we count the sign changes in \( f(-x) = x^4 + 2x^3 + 8x + 16 \):
  - There are **no sign changes** in \( f(-x) \), indicating **0 negative real zeros**.

### d) Find all the real zeros of the polynomial function. State any multiplicity.

We will attempt to find the real zeros by using the Rational Root Theorem and synthetic division or factoring.

The possible rational roots are the factors of the constant term (16) divided by the factors of the leading coefficient (1). Therefore, the possible rational roots are:

\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm 16
\]

Testing \( x = 2 \) using synthetic division:

\[
\begin{array}{r|rrrrr}
 2 & 1 & -2 & 0 & -8 & 16 \\
   &   & 2  & 0  & 0  & -16 \\
\hline
   & 1 & 0  & 0  & -8 & 0 \\
\end{array}
\]

Since the remainder is 0, \( x = 2 \) is a root.

Thus, we can factor \( f(x) \) as:

\[
f(x) = (x - 2)(x^3 - 8)
\]

Now, factor \( x^3 - 8 \) as a difference of cubes:

\[
f(x) = (x - 2)(x - 2)(x^2 + 2x + 4)
\]

Thus, the real zero is \( x = 2 \) with multiplicity 2. The quadratic factor \( x^2 + 2x + 4 \) has no real solutions, as its discriminant \( (2^2 - 4 \cdot 1 \cdot 4 = -12) \) is negative.

Therefore, the only real zero is \( x = 2 \), with multiplicity 2.

### e) Sketch the graph
- The polynomial has a real zero at \( x = 2 \), with multiplicity 2, meaning the graph touches the x-axis at \( x = 2 \) and turns around.
- Since the leading coefficient is positive, the graph rises towards infinity on both ends.
- The maximum number of turning points is 3.

The graph will rise from \( -\infty \), touch the x-axis at \( x = 2 \), and rise back towards \( \infty \).

### Summary of the Real Zeros:
- \( x = 2 \), multiplicity 2.

Would you like further details or explanations on any part of this process? Here are 5 additional questions related to the topic:

1. How does the multiplicity of a root affect the behavior of the graph?
2. Can you apply the Rational Root Theorem to more complex polynomials?
3. How does Descartes' Rule of Signs work for polynomials of higher degrees?
4. How do you find imaginary roots for polynomials like \( x^2 + 2x + 4 \)?
5. How can synthetic division help in simplifying polynomial expressions?

**Tip**: When testing rational roots, always consider both positive and negative values to capture all possible roots efficiently!

Given the polynomial function: f(x) = x^4 - 2x^3 - 8x + 16. a) Determine the number of turning points. b) Describe the end behavior. c) Use Descartes' Rule of Signs to determine the number of positive and negative real zeros. d) Find all the real zeros of the polynomial function. State any multiplicity. e) Sketch the graph.

Explore how to analyze and solve the polynomial function f(x) = x^4 - 2x^3 - 8x + 16. This step-by-step guide covers key topics like determining turning points, end behavior, Descartes' Rule of Signs, and finding real zeros with multiplicity. Suitable for high school students in Grades 10-12.

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