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

Short answer

No real zeros. No x-axis crossing or touching. Up to 5 turning points. Resembles \(-2x^6\).

Step-by-step solution

- Factor the Polynomial

Write the polynomial in its factorized form. Note that the given polynomial is already in a factorized form:\[ f(x) = -2(x^2 + 3)^3 \]

- Find Real Zeros and Their Multiplicity

Set the factorized polynomial equal to zero to find the real zeros:\[ -2(x^2 + 3)^3 = 0 \]Since \( x^2 + 3 = 0 \) has no real solutions, there are no real zeros for this polynomial.

- Determine Graph Behavior at Real Zeros

Since there are no real zeros, the graph neither crosses nor touches the x-axis at any point.

- Determine the Maximum Number of Turning Points

The degree of the polynomial is the exponent when expanded. The term \((x^2 + 3)^3\) expanded has a degree of 6 (since \(2 \times 3 = 6\)). The maximum number of turning points is given by the degree minus 1:\[ 6 - 1 = 5 \]Hence, there can be up to 5 turning points.

- Determine the End Behavior

For large values of \(|x|\), the polynomial resembles its leading term when the polynomial is expanded. The highest power of the polynomial is found by considering the term with the highest power when expanded:\[ -2(x^2)^3 = -2x^6 \]Therefore, the end behavior is similar to the power function:\[ f(x) \text{ behaves like } -2x^6 \text{ for large values of } |x| \]

Key concepts

Real Zeros

Real zeros of a polynomial are the values of x for which the polynomial equals zero. These are crucial as they tell us the x-intercepts of the polynomial's graph.In the given function
\( f(x) = -2(x^2 + 3)^3 \)
To find the real zeros:
Set \( f(x) = 0 \)
\[ -2(x^2 + 3)^3 = 0 \]
Since \( x^2 + 3 = 0 \) has no real solutions (because you can't have a negative number inside a square root), this means there are no real zeros.
Simply put, the graph of this polynomial never crosses the x-axis.

Multiplicity

Multiplicity of a root tells us how many times a particular zero appears in the polynomial. It reflects on the graph's behavior near that root. For instance, if the zero \( a \) appears twice (like \((x - a)^2)\) , it has a multiplicity of 2.
In our polynomial
\( (x^2 + 3)^3 \)
each term is raised to the third power, but since there are no real zeros, there's no real multiplicity to discuss here. If there were a real zero and it had a multiplicity, it would influence whether the graph touches or crosses the x-axis at that zero.

Turning Points

Turning points are where the graph changes direction from increasing to decreasing or vice versa. For a polynomial of degree n, the maximum number of turning points is \( n-1 \).
For the given polynomial:
\( f(x) = -2(x^2 + 3)^3 \)
Expanded, this is a 6th-degree polynomial ( \( 2 \times 3 = 6 \)).
Therefore, it can have up to 5 turning points (\( 6 - 1 = 5 \)).
Thus, there could be as many as five changes in direction on the graph.

End Behavior

End behavior describes how the polynomial behaves as x approaches large positive or negative values. To determine this, we focus on the leading term of the expanded polynomial. For
\( f(x) = -2(x^2 + 3)^3 \)
The term with the highest power is \( -2x^6 \). For large values of \(|x|\), the polynomial will resemble \( -2x^6 \).
This means:

• As \( x \rightarrow \text{+}∞\), \( f(x) \rightarrow -∞ \)
• As \( x \rightarrow -∞\), \( f(x) \rightarrow -∞ \)

Both ends of the graph will point downwards due to the negative leading coefficient.