B1. From the graph, read off the zeros and the initial value of the function and write down the intervals where the function values are negative. Write the equation of a fourth-degree polynomial whose graph is shown in the figure.
B1. From the graph, we read the zeros of the polynomial $x_{1}=-1, x_{2,3}=1$ and $x_{4}=4$ and the initial value $p(0)=-1$. The intervals where the function is negative are $(-1,1)$ and $(1,4)$. To write the equation of the fourth-degree polynomial, we use the form of the polynomial for the zeros $p(x)=a(x+1)(x-1)^{2}(x-4)$. Considering that $p(0)=-1$, we calculate the leading coefficient $a=\frac{1}{4}$. Thus, we have $p(x)=\frac{1}{4}(x+1)(x-1)^{2}(x-4)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-15.jpg?height=57&width=1637&top_left_y=2336&top_left_x=278)

Initial value of the polynomial read from the graph: $p(0)=-1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Intervals where the polynomial has a negative value: $(-1,1) \cup(1,4) \ldots .1$ point

Using the form of the polynomial for the zeros: $p(x)=a(x+1)(x-1)^{2}(x-4) \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-15.jpg?height=60&width=1641&top_left_y=2534&top_left_x=276)

Writing the polynomial.................................................................................................