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 \(|x|\). $$ f(x)=(x+\sqrt{3})^{2}(x-2)^{4} $$

Short answer

Zeros: \( x = -\sqrt{3} \) (multiplicity 2), \( x = 2 \) (multiplicity 4). Touches the x-axis at both zeros. Maximum turning points: 5. End behavior: resembles \( x^6 \).

Step-by-step solution

- Identify real zeros

Set the function equal to zero and solve for the values of x: \( (x + \sqrt{3})^2 (x - 2)^4 = 0 \). This gives us the zeros: \( x = -\sqrt{3} \) and \( x = 2 \).

- Find the multiplicity of each zero

The multiplicity is the exponent of each factor: For \( x = -\sqrt{3} \), the multiplicity is 2. For \( x = 2 \), the multiplicity is 4.

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

If the multiplicity of a zero is odd, the graph crosses the x-axis at that zero. If the multiplicity is even, the graph touches the x-axis: \( x = -\sqrt{3} \): Multiplicity is 2, so the graph touches the x-axis. \( x = 2 \): Multiplicity is 4, so the graph also touches the x-axis.

- Determine the maximum number of turning points

The maximum number of turning points is one less than the degree of the polynomial: The degree is 6 (2 from \( (x+\sqrt{3})^2 \) and 4 from \( (x-2)^4 \)). So, the maximum number of turning points is 6 - 1 = 5.

- Determine the end behavior

For large values of \( |x| \), the term with the highest degree dominates the function. The highest degree term is \( (x+\sqrt{3})^2 (x-2)^4 \approx x^6 \). Thus, the function behaves like \( f(x) \approx x^6 \) for large \( |x| \).

Key concepts

Real Zeros

When studying polynomial functions, real zeros are the solutions where the function equals zero. For the polynomial function given, we set it equal to zero to find the zeros:

\[(x + \sqrt{3})^2(x - 2)^4 = 0 \].

Solving this equation gives the zeros of the function:

• \(x = -\sqrt{3}\)
• \(x = 2\)

Each zero represents a point where the graph either touches or crosses the x-axis. Real zeros are essential to understanding the behavior of polynomial functions on a graph.

Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears in the polynomial function. It is indicated by the exponent on the factored term:

• For \(x = -\sqrt{3}\), the term is \((x + \sqrt{3})^2\), so the multiplicity is 2.
• For \(x = 2\), the term is \((x - 2)^4\), so the multiplicity is 4.

The multiplicity of a zero helps determine whether the graph of the function will cross the x-axis (if the multiplicity is odd) or just touch the x-axis (if the multiplicity is even). Understanding multiplicity gives insight into the function's interactions at its zeros.

X-Intercepts

X-intercepts are the points where the graph crosses or touches the x-axis. These points correspond to the real zeros of the function. For the given polynomial function:

• At \(x = -\sqrt{3}\), the multiplicity is 2, so the graph touches the x-axis.
• At \(x = 2\), the multiplicity is 4, so the graph also touches the x-axis.

This behavior is crucial in sketching the graph accurately. Knowing whether the graph touches or crosses the x-axis allows us to predict the behavior of the function around its zeros.

Turning Points

Turning points represent the points on the graph where the function changes its direction from increasing to decreasing or vice versa. The maximum number of turning points for any polynomial function is one less than its degree. For our function with a degree of 6:

\[\text{Maximum turning points} = 6 - 1 = 5\]

So, the graph can have up to five turning points. Observing turning points helps in understanding the peaks and troughs of the polynomial function, adding depth to its graphical representation.

End Behavior

End behavior describes what happens to the values of the polynomial function as \(x\) approaches very large positive or negative values. For our polynomial function, the term with the highest degree dictates this behavior, which is\((x + \sqrt{3})^2 (x - 2)^4 \approx x^6\).

Therefore, our polynomial function's end behavior resembles that of \(f(x) \approx x^6\). As \(x\) approaches infinity or negative infinity:

• If the highest degree term has an even exponent, the graph rises to positive infinity on both ends.

Understanding end behavior helps in predicting the general shape of the graph as \(x\) grows larger or smaller to give a comprehensive visual of the polynomial's behavior.