Question

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

answer attempt 1 out of 2
the degree of $f(x)$ is $\quad$ and the leading coefficient is $\quad$. there are $\square$ different real zeros and $\square$ relative extremes.

Explanation:

Step1: Determine the degree of the polynomial

The end - behavior of a polynomial is determined by the degree (highest power of \(x\)) and the leading coefficient. For a polynomial, if the degree \(n\) is odd: as \(x ightarrow+\infty\), if the leading coefficient \(a_n>0\), \(f(x) ightarrow+\infty\); if \(a_n < 0\), \(f(x) ightarrow-\infty\). As \(x ightarrow-\infty\), if \(a_n>0\), \(f(x) ightarrow-\infty\); if \(a_n < 0\), \(f(x) ightarrow+\infty\). If the degree \(n\) is even: as \(x ightarrow\pm\infty\), \(f(x)\) has the same sign (both \(+\infty\) or both \(-\infty\)) depending on the leading coefficient.

Looking at the graph, as \(x ightarrow+\infty\), \(f(x) ightarrow+\infty\) and as \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\). So the degree of the polynomial is odd. Also, the number of turning points (relative extrema) of a polynomial is at most \(n - 1\), where \(n\) is the degree of the polynomial.

Step2: Determine the leading coefficient sign

Since as \(x ightarrow+\infty\), \(f(x) ightarrow+\infty\) and the degree is odd, the leading coefficient is positive.

Step3: Count the number of real zeros

A real zero of a polynomial is a point where the graph intersects the \(x\) - axis. From the graph, we can see that the graph intersects the \(x\) - axis at 4 points? Wait, no. Wait, let's look again. Wait, there is a touch - and - turn (a repeated zero) on the left, then it crosses the \(x\) - axis, then crosses again, then crosses again? Wait, no. Wait, the graph: first, it touches the \(x\) - axis (a repeated root, so a zero with even multiplicity), then crosses the \(x\) - axis, then crosses again, then crosses again? Wait, no. Wait, the number of distinct real zeros: let's count the number of times the graph intersects or touches the \(x\) - axis. The graph touches the \(x\) - axis at one point (a repeated zero) and crosses the \(x\) - axis at 3 other points? Wait, no, looking at the graph: the left - most part touches the \(x\) - axis (so that's one zero, with multiplicity at least 2), then crosses the \(x\) - axis, then crosses again, then crosses again? Wait, no, the graph as shown: let's count the \(x\) - intercepts. The graph intersects the \(x\) - axis at 4 points? Wait, no, the left - most is a touch (so a zero with even multiplicity), then three crossings? Wait, no, the graph: starting from the left, it comes up, touches the \(x\) - axis (so that's a zero), then goes down, crosses the \(y\) - axis, then goes up, crosses the \(x\) - axis, then goes down, crosses the \(x\) - axis, then goes up. Wait, actually, the number of distinct real zeros: the graph touches the \(x\) - axis at 1 point (a repeated zero) and crosses the \(x\) - axis at 3 points? No, wait, the number of times the graph intersects the \(x\) - axis (including touches) is 4? Wait, no, let's count again. The left - most: touches the \(x\) - axis (1 zero), then crosses the \(x\) - axis at 3 other points? Wait, no, the graph in the picture: the left - hand side touches the \(x\) - axis (so that's one zero, multiplicity at least 2), then crosses the \(x\) - axis, then crosses again, then crosses again? Wait, no, the graph has 4 \(x\) - intercepts? Wait, no, the left - most is a touch (so that's a zero), then three crossings? Wait, no, the number of distinct real zeros: when we count distinct real zeros, a touch (a zero with even multiplicity) is still one distinct zero, and each crossing is a distinct zero. So from the graph, the number of distinct real zeros: let's see, the graph touches the \(x\) - axis at 1 point (so 1 zero) and crosses the \(x\) - axis at 3 points? Wait, no, looking at the graph, the number of times the graph intersects the \(x\) - axis (including the touch) is 4? Wait, no, the left - most is a touch (1 zero), then three crossings, so total 4 distinct real zeros? Wait, no, the touch is one zero (with multiplicity at least 2), and then three other zeros, so total 4 distinct real zeros? Wait, maybe I made a mistake. Wait, the degree of the polynomial: the number of turning points (relative extrema) is the number of times the graph changes direction. From the graph, we can see that there are 4 turning points? Wait, no, let's count the turning points. The graph goes up, then down, then up, then down, then up. So the number of turning points (relative extrema) is 4. Since the number of turning points of a polynomial is at most \(n - 1\), where \(n\) is the degree. So if there are 4 turning points, then \(n-1\geq4\), so \(n\geq5\). And since the end - behavior is odd (as \(x ightarrow+\infty\), \(f(x) ightarrow+\infty\); \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\)), the degree \(n\) is odd. So \(n = 5\) (since \(n-1 = 4\) turning points, so \(n=5\)).

Step4: Determine the leading coefficient sign

Since as \(x ightarrow+\infty\), \(f(x) ightarrow+\infty\) and the degree \(n = 5\) (odd), the leading coefficient is positive.

Step5: Count the number of real zeros

The graph intersects the \(x\) - axis at 4 points? Wait, no, the left - most is a touch (a zero with even multiplicity) and three crossings. Wait, the number of distinct real zeros: the touch is one zero, and the three crossings are three more zeros, so total 4 distinct real zeros? Wait, no, the left - most touch: when the graph touches the \(x\) - axis and turns around, that's a zero with even multiplicity (at least 2). Then the graph crosses the \(x\) - axis three times. So the number of distinct real zeros is 4? Wait, no, let's look at the graph again. The graph: starts from the bottom left, goes up, touches the \(x\) - axis (so that's a zero), then goes down, crosses the \(y\) - axis, then goes up, crosses the \(x\) - axis, then goes down, crosses the \(x\) - axis, then goes up. Wait, so the number of times it intersects the \(x\) - axis: 1 (touch) + 3 (crosses) = 4? Wait, no, the touch is one point, and then three crossing points, so total 4 real zeros (distinct? Wait, the touch is a distinct zero, and the three crossings are distinct zeros, so 4 distinct real zeros? Wait, no, maybe the touch is a repeated zero (multiplicity 2) and the other three are simple zeros, so total number of real zeros (counting multiplicity) would be more, but the question says "different real zeros", so we count distinct ones. So the number of different real zeros is 4? Wait, no, looking at the graph, the left - most is a touch (so one zero), then the graph crosses the \(x\) - axis at three other points, so total 4 different real zeros? Wait, no, maybe I miscounted. Wait, the graph: let's count the \(x\) - intercepts. The first \(x\) - intercept (left - most) is a touch, then the graph crosses the \(x\) - axis at three more points. So the number of different real zeros is 4? Wait, no, the left - most touch: when \(x\) is some value, the graph touches the \(x\) - axis (so \(f(x)=0\) there), then the graph crosses the \(x\) - axis at three other \(x\) - values. So the number of different real zeros is 4? Wait, no, maybe the left - most is a single zero (with multiplicity 2) and the other three are single zeros, so total 4 different real zeros.

Step6: Count the number of relative extrema

The number of relative extrema (turning points) of a polynomial of degree \(n\) is at most \(n - 1\). For \(n = 5\), the number of turning points is at most 4. Looking at the graph, we can see that the graph has 4 turning points (relative extrema): it goes up, then down, then up, then down, then up. So the number of relative extrema is 4.

Answer:

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