Question

for each graph, list the zeros and the multiplicity, the degree of the polynomial, and write a possible equation.
4.
zeros and multiplicity:
degree
possible equation
5.
zeros and multiplicity:
degree
possible equation

Explanation:

Step1: Analyze Graph 4 (Top Graph)

- **Zeros and Multiplicity**: The graph touches the x - axis at \(x = - 1\) (since it has a turning point on the x - axis, indicating even multiplicity) and crosses the y - axis at \(x = 0\) (crosses the x - axis? Wait, looking at the graph, the x - intercept: the graph touches the x - axis at \(x=-1\) (so multiplicity even, let's say 2) and crosses at \(x = 0\)? Wait, no, the graph: let's re - examine. The top graph: when x approaches left, it's going up, touches x at \(x=-1\) (so a root with even multiplicity, say 2) and then crosses the y - axis? Wait, no, the x - intercepts: the graph touches the x - axis at \(x=-1\) (multiplicity 2) and maybe another? Wait, maybe I misread. Wait, the graph of a polynomial: the number of turning points and the end - behavior. End - behavior: as \(x ightarrow+\infty\), the graph goes down; as \(x ightarrow-\infty\), it goes up. So the leading coefficient is negative, and the degree is odd? Wait, no, end - behavior: if degree is odd, leading coefficient negative: as \(x ightarrow+\infty\), \(y ightarrow-\infty\); as \(x ightarrow-\infty\), \(y ightarrow+\infty\). The number of turning points: the graph has 2 turning points, so degree at least 3. Wait, the x - intercepts: the graph touches the x - axis at \(x=-1\) (multiplicity 2, since it touches, not crosses) and crosses at \(x = 0\)? Wait, no, the graph: let's see, the top graph: when x = - 1, it touches the x - axis (so root \(x=-1\), multiplicity 2), and when x = 0, it crosses? Wait, no, the y - axis is at x = 0. Wait, maybe the x - intercepts are \(x=-1\) (multiplicity 2) and \(x = 0\) (multiplicity 1)? Wait, no, the graph: the top graph has a touch at \(x=-1\) (even multiplicity) and a cross at \(x = 0\) (odd multiplicity). So zeros: \(x=-1\) (multiplicity 2), \(x = 0\) (multiplicity 1).
- **Degree**: The number of turning points is 2. The degree of a polynomial is at least one more than the number of turning points. So degree at least 3. Since we have a root of multiplicity 2 and a root of multiplicity 1, total degree \(2 + 1=3\)? Wait, no, if \(x=-1\) has multiplicity 2 and \(x = 0\) has multiplicity 1, degree is \(2 + 1=3\). But the end - behavior: for degree 3, leading coefficient negative (since as \(x ightarrow+\infty\), \(y ightarrow-\infty\); \(x ightarrow-\infty\), \(y ightarrow+\infty\)), which matches.
- **Possible Equation**: Using the roots, \(y=-(x + 1)^2x=-x(x^{2}+2x + 1)=-x^{3}-2x^{2}-x\)

Step2: Analyze Graph 5 (Bottom Graph)

- **Zeros and Multiplicity**: The graph crosses the x - axis at \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 1), and \(x = 2\) (multiplicity 1)? Wait, no, the bottom graph: end - behavior: as \(x ightarrow+\infty\), \(y ightarrow+\infty\); as \(x ightarrow-\infty\), \(y ightarrow+\infty\), so degree is even, leading coefficient positive. The number of turning points: 3, so degree at least 4. The x - intercepts: crosses at \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 1), \(x = 2\) (multiplicity 1)? Wait, no, the graph: let's see, it crosses the x - axis three times? Wait, the bottom graph: as \(x ightarrow+\infty\), up; \(x ightarrow-\infty\), up. So degree even. Number of turning points: 3, so degree at least 4. The x - intercepts: let's say \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 1), \(x = 2\) (multiplicity 1). Then degree is \(1 + 1+1=3\)? No, that's odd. Wait, maybe one of the roots has even multiplicity. Wait, the graph: maybe \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 2), \(x = 2\) (multiplicity 1). Then degree is \(1 + 2+1 = 4\) (even), which matches the end - behavior (both ends up, leading coefficient positive).
- **Degree**: 4 (since end - behavior is even degree, leading coefficient positive, and number of turning points 3, degree at least 4)
- **Possible Equation**: \(y=(x + 3)x^{2}(x - 2)=x^{2}(x + 3)(x - 2)=x^{2}(x^{2}+x - 6)=x^{4}+x^{3}-6x^{2}\)

For Graph 4:

- **Zeros and Multiplicity**: \(x=-1\) (multiplicity 2), \(x = 0\) (multiplicity 1)
- **Degree**: 3
- **Possible Equation**: \(y=-x(x + 1)^{2}\) (or equivalent, e.g., \(y=-x^{3}-2x^{2}-x\))

For Graph 5:

- **Zeros and Multiplicity**: \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 2), \(x = 2\) (multiplicity 1)
- **Degree**: 4
- **Possible Equation**: \(y=x^{2}(x + 3)(x - 2)\) (or equivalent, e.g., \(y=x^{4}+x^{3}-6x^{2}\))

Answer:

Graph 4:

• Zeros and Multiplicity: \(x=-1\) (multiplicity 2), \(x = 0\) (multiplicity 1)
• Degree: 3
• Possible Equation: \(y=-x(x + 1)^{2}\)

Graph 5:

• Zeros and Multiplicity: \(x=-3\) (multiplicity 1), \(x = 0\) (multiplicity 2), \(x = 2\) (multiplicity 1)
• Degree: 4
• Possible Equation: \(y=x^{2}(x + 3)(x - 2)\)