Question

for the polynomial function f(x) = 4(x - 6)(x + 6)² answer the following questions.
(a) list each real zero and its multiplicity.
(b) determine whether the graph crosses or touches the x-axis at each x-intercept.
(c) determine the maximum number of turning points on the graph.
(d) determine the end behavior, that is, find the power function that the graph of f resembles for large values of |x|
(a) find any real zeros of f. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the real zero of f is \\(\square\\) with multiplicity \\(\square\\)
(type an exact answer, using radicals as needed. type integers or fractions)

b. the smallest zero of f is \\(\square\\) with multiplicity \\(\square\\). the largest zero of f is \\(\square\\) with multiplicity \\(\square\\)
(type an exact answer, using radicals as needed. type integers or fractions)

c. the smallest zero of f is \\(\square\\) with multiplicity \\(\square\\). the middle zero of f is \\(\square\\) with multiplicity \\(\square\\). the largest zero of f is \\(\square\\) with multiplicity \\(\square\\)
(type an exact answer, using radicals as needed. type integers or fractions)

d. there are no real zeros.
(b) select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the graph crosses the x-axis at \\(\square\\)
(type an exact answer, using radicals as needed. type an integer or a simplified fraction. use a comma to separate answers as needed.)

b. the graph touches the x-axis at \\(\square\\)
(type an exact answer, using radicals as needed. type an integer or a simplified fraction. use a comma to separate answers as needed.)

c. the graph touches the x-axis at \\(\square\\) and crosses at \\(\square\\)
(type an exact answer, using radicals as needed. type integers or simplified fractions. use a comma to separate answers as needed.)

d. the graph neither crosses nor touches the x-axis.
(c) the maximum number of turning points on the graph is \\(\square\\)
(type a whole number.)

(d) the power function that the graph of f resembles for large values of |x| is y = \\(\square\\)

Explanation:

Step1: Find Real Zeros

To find the real zeros of the polynomial \( f(x) = 4(x - 6)(x + 6)^2 \), we set \( f(x) = 0 \). Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So we have:
\( 4(x - 6)(x + 6)^2=0 \)
Since \( 4
eq0 \), we set \( x - 6 = 0 \) or \( (x + 6)^2=0 \)
For \( x - 6 = 0 \), we get \( x = 6 \). For \( (x + 6)^2=0 \), we get \( x=-6 \) (with multiplicity 2).

Step2: Analyze Multiplicity and Graph Behavior at Zeros

• For the zero \( x = 6 \): The factor \( (x - 6) \) has an exponent of 1 (odd multiplicity). So the graph of the function crosses the x - axis at \( x = 6 \).
• For the zero \( x=-6 \): The factor \( (x + 6) \) has an exponent of 2 (even multiplicity). So the graph of the function touches the x - axis at \( x=-6 \) and then turns around.

Step3: Find the Smallest and Largest Zeros with Multiplicity

• The zeros are \( x=-6 \) (multiplicity 2) and \( x = 6 \) (multiplicity 1). The smallest zero is \( x=-6 \) with multiplicity 2. The largest zero is \( x = 6 \) with multiplicity 1.

Step4: Determine the Maximum Number of Turning Points

The degree of the polynomial \( f(x)=4(x - 6)(x + 6)^2=4(x - 6)(x^{2}+12x + 36)=4(x^{3}+12x^{2}+36x-6x^{2}-72x - 216)=4(x^{3}+6x^{2}-36x - 216)=4x^{3}+24x^{2}-144x - 864 \). The degree \( n = 3 \). The maximum number of turning points of a polynomial function of degree \( n \) is \( n - 1 \). So for \( n = 3 \), the maximum number of turning points is \( 3-1=2 \).

Step5: Determine the End - Behavior Power Function

For large values of \(|x|\), the leading term of the polynomial \( f(x)=4x^{3}+24x^{2}-144x - 864 \) determines the end - behavior. The leading term is \( 4x^{3} \). So the power function that the graph of \( f \) resembles for large values of \(|x|\) is \( y = 4x^{3}\), or we can say the leading - term is \( x^{3}\) (the coefficient 4 affects the vertical stretch but the end - behavior is dominated by the power of \( x \), so the power function is \( y=x^{3}\) in terms of the power of \( x \)).

Answer:

s:
(a) The real zeros are \( x = 6 \) (multiplicity 1) and \( x=-6 \) (multiplicity 2).
(b) A. The graph crosses the x - axis at \( x = 6 \); B. The graph touches the x - axis at \( x=-6 \)
The smallest zero of \( f \) is \( x=-6 \) with multiplicity \( 2 \); The largest zero of \( f \) is \( x = 6 \) with multiplicity \( 1 \)
(c) The maximum number of turning points is \( 2 \)
(d) The power function is \( y = x^{3}\) (or \( y = 4x^{3}\))