Question

What information must you have to write a polynomial function having all real roots given its graph?

Answer 1

Items to include in the response are:

1. the x-intercepts of the graph, or the zeros of the function

2. the multiplicity of its factors

3. a point on the graph that is not a root, or the value of the leading coefficient

Answer 2

the roots are simple

ok, so
remember some nice rules


if the zeores of the function (where the graph hits the x axis) are r1,r2...rn the the factors are (x-r1)(x-r2)...(x-rn)

if the degree (how many factors, or x^degree) is odd, then the ends of the graph go in oposite directions, from bottom left to top rght
if the leading coefient (first term) has a negative like f(x)=-3(x-r1)(x-r2)...
then the graph is reflected across the x axis

if the degree is even, the ends go in same direction
if leading coefient is negative, then ends both go down


the roots are wehre the graph hits te x axis
if yo hav a root repeat, that's called a multiplicaty
(x-r1)^2 is even multiplicy
(x-r1)^3 is odd
odd multilicity means the line goes though the graph
even means that the graph just touches the graph at that point and bounces off


this is confusing so sorry