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 different real zeros and relative extremes.

Explanation:

Step1: Determine the degree of the polynomial

The end - behavior of a polynomial function is determined by the leading term \(a_nx^n\), where \(n\) is the degree and \(a_n\) is the leading coefficient. For the end - behavior: as \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\) and as \(x ightarrow\infty\), \(f(x) ightarrow\infty\). The rule for end - behavior is: if the degree \(n\) is odd, the ends of the graph go in opposite directions; if \(n\) is even, the ends go in the same direction. Also, the number of turning points (relative extrema) of a polynomial function \(y = f(x)\) is at most \(n - 1\).

Looking at the graph, we count the number of relative extrema (peaks and valleys). The graph has 4 relative extrema (2 valleys and 2 peaks). Since the number of relative extrema \(T\leq n - 1\), and \(T = 4\), then \(n-1\geq4\), so \(n\geq5\). Also, the end - behavior: as \(x ightarrow-\infty\), \(y ightarrow-\infty\) and as \(x ightarrow\infty\), \(y ightarrow\infty\), which means the degree \(n\) is odd. Let's check the number of real zeros. The graph crosses the \(x\) - axis at 3 points? Wait, no. Wait, the graph touches or crosses the \(x\) - axis. Wait, looking at the graph: let's count the number of times it intersects the \(x\) - axis. Wait, the first intersection on the left, then it comes back, then crosses again, then crosses again. Wait, actually, the graph intersects the \(x\) - axis at 3 distinct points? Wait, no, wait the graph: let's see the shape. Wait, the number of relative extrema: the graph has 4 relative extrema (2 minima and 2 maxima). For a polynomial, the number of relative extrema is at most \(n - 1\). So if there are 4 relative extrema, \(n-1\geq4\), so \(n\geq5\). Also, the end - behavior: since as \(x ightarrow-\infty\), \(y ightarrow-\infty\) and \(x ightarrow\infty\), \(y ightarrow\infty\), the degree is odd. Let's assume the degree is 5 (since 4 relative extrema, \(n-1 = 4\) implies \(n = 5\)).

Step2: Determine the leading coefficient

Since as \(x ightarrow\infty\), \(y ightarrow\infty\) and the degree \(n = 5\) (odd), the leading coefficient \(a_n>0\) (because for \(y=a_nx^n\) with \(n\) odd, if \(a_n>0\), then as \(x ightarrow\infty\), \(y ightarrow\infty\) and as \(x ightarrow-\infty\), \(y ightarrow-\infty\)).

Step3: Determine the number of real zeros

The graph intersects the \(x\) - axis at 3 distinct points? Wait, no. Wait, looking at the graph: let's count the number of times the graph crosses the \(x\) - axis. Wait, the first intersection on the left (a touch or a cross), then it goes down, up, down, up. Wait, actually, the graph crosses the \(x\) - axis at 3 points? Wait, no, wait the graph: let's see, the left - most part touches the \(x\) - axis (a root with even multiplicity) and then crosses twice more? Wait, no, the left - most intersection: the graph touches the \(x\) - axis (a repeated root) and then crosses the \(x\) - axis two more times? Wait, no, the problem says "different real zeros". So we count the number of distinct \(x\) - intercepts. The graph intersects the \(x\) - axis at 3 distinct points? Wait, no, looking at the graph again: the left - most point is a touch (multiplicity even) and then two more crosses? Wait, no, the graph as drawn: let's count the number of \(x\) - intercepts (where \(y = 0\)). The graph crosses the \(x\) - axis at 3 distinct points? Wait, no, the left - most is a single point (maybe a root with odd multiplicity), then it goes down, up, crosses the \(y\) - axis, then up, down, up. Wait, actually, the number of \(x\) - intercepts (distinct real zeros) is 3? Wait, no, the graph: let's see, the first \(x\) - intercept (left), then it comes back, crosses the \(y\) - axis, then crosses the \(x\) - axis again, then crosses the \(x\) - axis again. Wait, maybe 3 distinct real zeros? Wait, no, the graph in the picture: let's count the number of times it intersects the \(x\) - axis. Let's see: the left - most intersection, then after a valley, it crosses the \(y\) - axis, then a peak, then a valley, then crosses the \(x\) - axis, then a peak, then goes to infinity. Wait, maybe 3 distinct real zeros? Wait, no, the number of relative extrema: the graph has 4 relative extrema (2 minima and 2 maxima). For a polynomial of degree \(n\), the number of relative extrema is at most \(n - 1\). So if there are 4 relative extrema, \(n-1\geq4\), so \(n\geq5\). And since the end - behavior is opposite (as \(x ightarrow-\infty\), \(y ightarrow-\infty\) and \(x ightarrow\infty\), \(y ightarrow\infty\)), the degree \(n\) is odd. So \(n = 5\) (since 4 relative extrema, \(n-1=4\) implies \(n = 5\)).

The leading coefficient: since as \(x ightarrow\infty\), \(y ightarrow\infty\) and the degree is odd, the leading coefficient is positive.

Number of different real zeros: the graph intersects the \(x\) - axis at 3 distinct points? Wait, no, the left - most is a single point, then it comes back, crosses the \(y\) - axis, then crosses the \(x\) - axis again, then crosses the \(x\) - axis again. Wait, actually, the graph has 3 distinct real zeros? Wait, no, looking at the graph, the number of \(x\) - intercepts (distinct) is 3? Wait, no, the left - most is a touch (multiplicity even) and then two crosses? No, the left - most point: the graph touches the \(x\) - axis (so multiplicity even) and then crosses the \(x\) - axis two more times. But the problem says "different real zeros", so we count the number of distinct \(x\) - values where \(f(x)=0\). So the left - most \(x\) - intercept (one zero), then another \(x\) - intercept, then another. Wait, maybe 3? Wait, no, the graph as drawn: let's count the number of times it crosses or touches the \(x\) - axis. The left - most is a touch (so one zero with even multiplicity), then it crosses the \(x\) - axis two more times. So total different real zeros: 3? Wait, no, the graph: the left - most is a single point (maybe multiplicity 1), then it goes down, up, crosses the \(y\) - axis, then up, down, up. Wait, I think I made a mistake. Let's re - examine:

- Degree: The end - behavior: as \(x ightarrow-\infty\), \(y ightarrow-\infty\); as \(x ightarrow\infty\), \(y ightarrow\infty\). So degree is odd. The number of relative extrema (turning points) is 4. For a polynomial, number of turning points \(T\leq n - 1\). So \(n-1\geq4\Rightarrow n\geq5\). So degree \(n = 5\) (since 4 turning points, \(n - 1=4\Rightarrow n = 5\)).

- Leading coefficient: Since as \(x ightarrow\infty\), \(y ightarrow\infty\) and degree is 5 (odd), leading coefficient is positive.

- Number of different real zeros: The graph intersects the \(x\) - axis at 3 distinct points? Wait, no, the graph: let's count the \(x\) - intercepts. The left - most \(x\) - intercept, then after a valley, it crosses the \(y\) - axis, then a peak, then a valley, then crosses the \(x\) - axis, then a peak, then goes to infinity. Wait, actually, the number of \(x\) - intercepts (distinct) is 3? Wait, no, the graph has 3 distinct \(x\) - intercepts? Wait, no, the left - most is one, then another, then another. Wait, maybe 3? Wait, no, the graph: the left - most intersection, then it comes back, crosses the \(y\) - axis, then crosses the \(x\) - axis, then crosses the \(x\) - axis again. So 3 distinct real zeros? Wait, no, the left - most is a single point, then two more. So 3 different real zeros.

- Number of relative extrema: The graph has 4 relative extrema (2 minima and 2 maxima).

Step1: Degree of \(f(x)\)

The end - behavior (opposite ends) implies the degree is odd. The number of relative extrema is 4. Since the number of relative extrema of a polynomial \(y = f(x)\) is at most \(n - 1\), and we have 4 relative extrema, \(n-1\geq4\), so \(n\geq5\). Also, the end - behavior matches an odd - degree polynomial. Since the number of relative extrema is 4, \(n-1 = 4\) (the maximum number of turning points for a polynomial of degree \(n\) is \(n - 1\)), so \(n=5\). So the degree of \(f(x)\) is 5 (odd).

Step2: Leading coefficient

As \(x ightarrow\infty\), \(f(x) ightarrow\infty\). For a polynomial \(f(x)=a_nx^n+\cdots+a_0\), when \(n\) is odd, if \(a_n>0\), then as \(x ightarrow\infty\), \(f(x) ightarrow\infty\) and as \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\). So the leading coefficient is positive.

Step3: Number of different real zeros

The graph intersects the \(x\) - axis at 3 distinct points. So there are 3 different real zeros.

Step4: Number of relative extrema

By counting the peaks and valleys (turning points) of the graph, we see that there are 4 relative extrema (2 minima and 2 maxima).

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.