Question

Polynomial function: (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 \(|x|\). $$ f(x)=-2 x^{2}\left(x^{2}-2\right) $$

Short answer

Zeros: 0 (touches), \(\pm \sqrt{2}\) (crosses). Max 3 turning points. End behavior: Like \(-2x^4\).

Step-by-step solution

- Expand the given function

First, expand the polynomial function to simplify it: \[ f(x) = -2x^2(x^2 - 2) \Rightarrow f(x) = -2x^4 + 4x^2 \]

- Find the real zeros

Set the function equal to zero and solve for the zeros: \[ -2x^4 + 4x^2 = 0 \Rightarrow -2x^2(x^2 - 2) = 0 \Rightarrow x^2(x^2 - 2) = 0 \Rightarrow x^2 = 0 \text{ or } x^2 = 2 \Rightarrow x = 0 \text{ or } x = \pm\sqrt{2} \]

- Determine the multiplicity of each zero

Analyze the multiplicity of each zero: - Zero at \(0\) has multiplicity 2 (from \(x^2\)).- Zeros at \(\pm \sqrt{2}\) each have multiplicity 1.

- Determine if the graph crosses or touches the x-axis

Analyze the behavior at each zero: - For zero at \(0\) with multiplicity 2, the graph touches the x-axis and turns around.- For zeros at \(\pm \sqrt{2}\) with multiplicity 1, the graph crosses the x-axis.

- Determine the maximum number of turning points

To find the maximum number of turning points, use the degree of the polynomial: \[ \text{Degree} = 4 \text{, so maximum number of turning points is} 4 - 1 = 3 \]

- Determine the end behavior

Examine the leading term to determine the end behavior: - Leading term is \(-2x^4\).- For large \(|x|\), the polynomial behaves like \(-2x^4\). Thus, as \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).

Key concepts

Real Zeros

Real zeros of a polynomial function are the values of x where the function equals zero. To find the real zeros of a polynomial, set the polynomial equation to zero and solve for x. In this exercise, the polynomial is given by:

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

To find its zeros, we solve:

\[ -2x^4 + 4x^2 = 0 \]

Factoring out the common term, we get:

\[ x^2(x^2 - 2) = 0 \]

This gives us:

\[ x^2 = 0 \text{ or } x^2 = 2 \]

Hence, the real zeros are:

• x = 0
• x = ±√2

These are the points where the graph intersects or touches the x-axis.

Multiplicity

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial equation. It affects how the graph behaves at the zeros.

• If the multiplicity is odd, the graph crosses the x-axis at that zero.
• If the multiplicity is even, the graph touches the x-axis and turns around at that zero.

For the given polynomial:

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

• The zero at x = 0 has multiplicity 2 (because of the term \(x^2\)).
• The zeros at x = ±√2 each have multiplicity 1 (each occurring once).

Turning Points

Turning points are where the graph changes direction from increasing to decreasing or vice versa. For any polynomial of degree n, the maximum number of turning points is n - 1.

For the polynomial:

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

The degree is 4. Therefore, the graph has a maximum of:

\[ 4 - 1 = 3 \text{ turning points} \]

This tells us that the graph can change direction up to three times.

End Behavior

End behavior describes how the graph behaves as x approaches very large positive or negative values. It is determined by the leading term of the polynomial.

For the polynomial:

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

The leading term is \(-2x^4\). Because the coefficient (−2) is negative and the exponent (4) is even, the end behavior is:

• As x → ∞, f(x) → -∞
• As x → -∞, f(x) → -∞

Graph Behavior

Graph behavior around the zeros is influenced by both the zeros themselves and their multiplicities.

• At x = 0 with multiplicity 2, the graph touches the x-axis and turns around.
• At x = ±√2 with multiplicity 1, the graph crosses the x-axis.

Examine the polynomial:

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

The polynomial's degree (4) and leading coefficient (−2) guide the overall graph shape: steep downward directions for large |x| values.

The graph's turning points and behavior at zeros anchor its detailed local shape.