Question

consider the following.\\( f(x) = x^3 - 2x^2 - 9x + 18 \\)\\( \text{(a) find all real zeros of the polynomial function. (enter your answers as a comma-separated list. if there is no solution, enter no solution.)} \\)\\( x = \\)\\( \text{(b) determine the multiplicity of each zero.} \\)\\( \text{smallest } x\text{-value} \\) \\( \text{---select---} \\)\\( \text{smallest } x\text{-value} \\) \\( \text{---select---} \\)\\( \text{largest } x\text{-value} \\) \\( \text{---select---} \\)\\( \text{(c) determine the maximum possible number of turning points of the graph of the function.} \\)\\( \text{turning point(s)} \\)\\( \text{(d) use a graphing utility to graph the function and verify your answers.}

Explanation:

Part (a)

Step1: Factor the polynomial

We have the function \( f(x) = x^3 - 2x^2 - 9x + 18 \). We can factor by grouping.
Group the terms: \( (x^3 - 2x^2) + (-9x + 18) \)
Factor out the common factors from each group: \( x^2(x - 2) - 9(x - 2) \)
Now, factor out \( (x - 2) \): \( (x - 2)(x^2 - 9) \)
Notice that \( x^2 - 9 \) is a difference of squares, so we can factor it further: \( (x - 2)(x - 3)(x + 3) \)

Step2: Find the zeros

To find the zeros, we set \( f(x) = 0 \):
\( (x - 2)(x - 3)(x + 3) = 0 \)
Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \), \( b = 0 \) or both.
So, \( x - 2 = 0 \) or \( x - 3 = 0 \) or \( x + 3 = 0 \)
Solving these equations:

• For \( x - 2 = 0 \), we get \( x = 2 \)
• For \( x - 3 = 0 \), we get \( x = 3 \)
• For \( x + 3 = 0 \), we get \( x=- 3 \)

Step1: Recall the multiplicity rule

For a polynomial \( f(x)=(x - a)^n \), the multiplicity of the zero \( x = a \) is \( n \). In our factored form \( f(x)=(x + 3)(x - 2)(x - 3) \), we can write each factor as \( (x-(-3))^1 \), \( (x - 2)^1 \) and \( (x - 3)^1 \)

Step2: Determine the multiplicity

For the zero \( x=-3 \) (from the factor \( (x + 3)=(x-(-3)) \)), the multiplicity is 1.
For the zero \( x = 2 \) (from the factor \( (x - 2) \)), the multiplicity is 1.
For the zero \( x = 3 \) (from the factor \( (x - 3) \)), the multiplicity is 1.

Since all multiplicities are 1 (odd), the graph of the function crosses the x - axis at each zero. And since the multiplicity of each zero is 1, for the smallest \( x \) - value (which is \( x=-3 \)) the multiplicity is 1, for the middle \( x \) - value (which is \( x = 2 \)) the multiplicity is 1, and for the largest \( x \) - value (which is \( x = 3 \)) the multiplicity is 1.

Step1: Recall the formula for the number of turning points

For a polynomial function of degree \( n \), the maximum number of turning points is \( n - 1 \).

Step2: Determine the degree of the polynomial

The function \( f(x)=x^3 - 2x^2 - 9x + 18 \) is a cubic polynomial, so the degree \( n = 3 \).

Step3: Calculate the maximum number of turning points

Using the formula \( n-1 \), with \( n = 3 \), we get \( 3 - 1=2 \). So the maximum possible number of turning points of the graph of the function is 2.

Answer:

\( -3, 2, 3 \)

Part (b)