Question

the polynomial function $f(x)$ is graphed below. fill in the form below regarding the features of this graph.

the degree of $f(x)$ is and the leading coefficient is . there are $\square$ different real zeros and $\square$ relative extremes.

Explanation:

Step1: Determine the degree (odd/even)

The end - behavior of a polynomial: as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\); as \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\). For a polynomial \(y = a_nx^n+\cdots+a_0\), if the leading coefficient \(a_n>0\) and the degree \(n\) is odd, the end - behavior is down on the left and up on the right. So the degree is odd. Also, the number of turning points (relative extrema) is related to the degree. The number of turning points of a polynomial of degree \(n\) is at most \(n - 1\). Here, we count the turning points.

Step2: Determine the leading coefficient sign

From the end - behavior (down on left, up on right), the leading coefficient is positive.

Step3: Count real zeros

The real zeros of a polynomial are the \(x\) - intercepts. Looking at the graph, we can see that the graph crosses the \(x\) - axis at 3 points, so there are 3 different real zeros.

Step4: Count relative extrema

Relative extrema are the local maxima and minima. By looking at the graph, we can see that there are 4 turning points (local max and min), so there are 4 relative extrema. Also, since the number of turning points is \(n - 1\), if \(n-1 = 4\), then \(n=5\) (so the degree is 5, which is odd).

Answer:

The degree of \(f(x)\) is \(\boldsymbol{5}\) (odd) and the leading coefficient is \(\boldsymbol{\text{positive}}\). There are \(\boldsymbol{3}\) different real zeros and \(\boldsymbol{4}\) relative extrema.