Let's walk through the problem:

Part a: Find the zeros and state their multiplicity

The graph shows the polynomial crossing or touching the x-axis at three points. These points are the zeros of the polynomial.

1. The zero at x = -1 touches the x-axis but does not cross it. This indicates that the multiplicity of the zero is even (in this case, 2). Hence, $x = -1$ has multiplicity 2.

2. The zero at x = 1 crosses the x-axis, which means it has an odd multiplicity (specifically 1).

3. The zero at x = 2 also crosses the x-axis, but the graph behaves in a way that shows a higher odd multiplicity (specifically 3).

• Therefore, the zeros are:
  • $x = -1$ with multiplicity 2 (even)
  • $x = 1$ with multiplicity 1 (odd)
  • $x = 2$ with multiplicity 3 (odd)

Part b: Write an equation for the polynomial

We need to write the polynomial as a product of factors. Each factor corresponds to a zero, and the multiplicity is the exponent on the factor.

• For $x = -1$, with multiplicity 2, the factor is $(x + 1)^2$.
• For $x = 1$, with multiplicity 1, the factor is $(x - 1)^1$.
• For $x = 2$, with multiplicity 3, the factor is $(x - 2)^3$.

The polynomial equation, with a leading coefficient of either 1 or -1, becomes:

$f(x) = (x + 1)^2(x - 1)(x - 2)^3$

Part c: Find the y-intercept

To find the y-intercept, substitute $x = 0$ into the equation from part (b):

$f(0) = (0 + 1)^2(0 - 1)(0 - 2)^3 = (1)^2(-1)(-2)^3 = 1 \times -1 \times (-8) = 8$

Thus, the y-intercept is 8.

Practice Problem:

Graph: Suppose a graph has the following behavior:

• A zero at $x = 0$ with multiplicity 2 (the graph touches the x-axis).
• A zero at $x = 3$ with multiplicity 1 (the graph crosses the x-axis).
• A zero at $x = -2$ with multiplicity 3 (the graph crosses the x-axis, but "flattens" out near this point).

Questions: a. Find the zeros and state their multiplicity.
b. Write an equation for the polynomial with a leading coefficient of 1.
c. Find the y-intercept.

Tip: Remember that the behavior of the graph at each zero can help determine its multiplicity. Zeros with even multiplicities cause the graph to touch the x-axis, while zeros with odd multiplicities cause it to cross the axis.

Do you have any questions or need more details? Here are some related questions for further practice:

1. How do you determine the multiplicity of a zero from a graph?
2. What is the effect of the leading coefficient on the graph's orientation?
3. How does the multiplicity of a zero affect the graph's behavior near that zero?
4. How can you determine the degree of a polynomial by analyzing its graph?
5. What is the significance of the y-intercept in a polynomial function?

Tip: Practice substituting different values into the polynomial equation to see how it behaves at various points!