Questions: For the polynomial function f(x)=-7 x^2(x^2-5) 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 with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) B. The smallest zero of f is with multiplicity . The largest zero of f is with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) C. The smallest zero of f is with multiplicity . The middle zero of f is with multiplicity The largest zero of f is with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) D. There are no real zeros.

For the polynomial function f(x)=-7 x^2(x^2-5) 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 with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) B. The smallest zero of f is with multiplicity . The largest zero of f is with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) C. The smallest zero of f is with multiplicity . The middle zero of f is with multiplicity The largest zero of f is with multiplicity . (Type an exact answer, using radicals as needed. Type integers or fractions.) D. There are no real zeros.
Transcript text: For the polynomial function $f(x)=-7 x^{2}\left(x^{2}-5\right)$ 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 $|\mathrm{x}|$. (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 integersor fractions.) D. There are no real zeros.
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Solution

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Solution Steps

To solve the given polynomial function \( f(x) = -7x^2(x^2 - 5) \), we need to address the following questions:

(a) List each real zero and its multiplicity:

  • Factor the polynomial to find the zeros.
  • Determine the multiplicity of each zero.

(b) Determine whether the graph crosses or touches the \( x \)-axis at each \( x \)-intercept:

  • Use the multiplicity of each zero to determine if the graph crosses or touches the \( x \)-axis.

(c) Determine the maximum number of turning points on the graph:

  • Use the degree of the polynomial to find the maximum number of turning points.
Solution Approach
  1. Factor the polynomial \( f(x) = -7x^2(x^2 - 5) \).
  2. Identify the zeros and their multiplicities.
  3. Use the multiplicities to determine if the graph crosses or touches the \( x \)-axis.
  4. Calculate the degree of the polynomial to find the maximum number of turning points.
Step 1: Factor the Polynomial

The polynomial function is given by

\[ f(x) = -7x^2(x^2 - 5). \]

This can be factored as

\[ f(x) = -7x^2(x - \sqrt{5})(x + \sqrt{5}). \]

Step 2: Identify the Real Zeros and Their Multiplicities

From the factored form, we can identify the real zeros:

  1. \( x = 0 \) with multiplicity \( 2 \)
  2. \( x = \sqrt{5} \) with multiplicity \( 1 \)
  3. \( x = -\sqrt{5} \) with multiplicity \( 1 \)
Step 3: Determine the Behavior at Each \( x \)-Intercept
  • For \( x = 0 \) (multiplicity \( 2 \)), the graph touches the \( x \)-axis.
  • For \( x = \sqrt{5} \) (multiplicity \( 1 \)), the graph crosses the \( x \)-axis.
  • For \( x = -\sqrt{5} \) (multiplicity \( 1 \)), the graph crosses the \( x \)-axis.
Step 4: Calculate the Maximum Number of Turning Points

The degree of the polynomial \( f(x) \) is \( 4 \). The maximum number of turning points is given by

\[ \text{Maximum Turning Points} = \text{Degree} - 1 = 4 - 1 = 3. \]

Final Answer

  • The real zeros and their multiplicities are:

    • \( x = 0 \) with multiplicity \( 2 \)
    • \( x = \sqrt{5} \) with multiplicity \( 1 \)
    • \( x = -\sqrt{5} \) with multiplicity \( 1 \)

  • The graph touches the \( x \)-axis at \( x = 0 \) and crosses at \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).

  • The maximum number of turning points is \( 3 \).

Thus, the answers are:

  • Zeros: \( (0, 2), (\sqrt{5}, 1), (-\sqrt{5}, 1) \)
  • Touches at \( x = 0 \), crosses at \( x = \sqrt{5} \) and \( x = -\sqrt{5} \)
  • Maximum turning points: \( 3 \)

\[ \boxed{\text{Zeros: } (0, 2), (\sqrt{5}, 1), (-\sqrt{5}, 1); \text{ Touches at } 0, \text{ crosses at } \sqrt{5}, -\sqrt{5}; \text{ Max turning points: } 3} \]

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