So I was given this graph:
and I was told the function was defined as:
$$f(x) = (x-c_1)(x-c_2)…(x-c_n)$$
Then I was told to find n. So I assumed they meant the total number of real roots. So since from the graph I could tell there were 2 real roots, then it was impossible for it to be 2 reals roots and 1 non-real root. So the answer must be 3 real roots. It was correct, but their explanation was:
The graph crosses the x‑axis at −10, so f must have a positive odd number of factors corresponding to that x‑intercept
The graph crosses the x‑axis at 15, so f must have a positive even number of factors corresponding to that x‑intercept
There are no other x‑intercepts, so this accounts for all the factors. The total number of factors of f is, therefore, an odd number plus an even number, and must be at least 3.
I don’t understand their explanation, because they first say f must have a positive odd number of factors, then they say it has positive even number of factors, and then they say there are no other x-intercepts? when there is another third factor? Does anyone know what is meant by that explanation?
