Answer:
Part 1)
The possible multiplicities are:
[tex]x=-7[/tex] multiplicity 1
[tex]x=-3[/tex] multiplicity 3
[tex]x=2[/tex] multiplicity 1
[tex]x=5[/tex] multiplicity 2
Part 2
The factored form is
[tex]p(x)=(x+7)(x+3)^3(x-2)(x-5)^2[/tex]
Step-by-step explanation:
Part 1.
The missing diagram is shown in the attachment.
The zeroes of the seventh degree polynomial are the x-intercepts of the graph.
From the graph, we have x-intercepts at:
[tex]x=-7[/tex], [tex]x=-3[/tex], [tex]x=2[/tex], and [tex]x=5[/tex].
The multiplicities tell us how many times a root repeats.
Also, even multiplicities will not cross their x-intercept, while odd multiplicities cross their x-intercepts.
The possible multiplicities are:
[tex]x=-7[/tex] multiplicity 1
[tex]x=-3[/tex] multiplicity 3
[tex]x=2[/tex] multiplicity 1
[tex]x=5[/tex] multiplicity 2
Note that the total multiplicity must equate the degree.
Part 2)
According to the factor theorem, if [tex]x=a[/tex] is a zero of p(x), then [tex](x-a)[/tex] is a factor.
Using the multiplicities , we can write the factors as:
[tex]x+7[/tex]
[tex](x+3)^3[/tex]
[tex](x-2)^1[/tex]
[tex](x-5)^2[/tex]
Therefore the completely factored form of this seventh degree polynomial is [tex]p(x)=(x+7)(x+3)^3(x-2)(x-5)^2[/tex]
