Question

//NOTE: This question has a graph included with points at (-2,0) (1,0) (2,0). all the zeros.

The polynomial P(x) polotted below is a cubic. From the polt below, it is easy to determine the three factors of P(x). With a little more work, you can also determine the leading coefficient of P(x).

(a) Find a factorization of P(x) which includes the unkown leading coefficient a and the three factors you can read from the plot, like P(x) =a(factor 1)(factor 2)(factor 3).

//NOTE: I ended up getting P(x) = a(x-1)(x+2)(x-2). I realized this could eventually be reverse factored all the way to a(x^3+4x-x^2-4). Not sure if that helps me in any way.

(b) Using the previous part and the fact that P(-1) goes through integer coordinates, set up an equation and solve for the unkown a.

   //NOTE: This is where I am lost. no idea what p(-1) goes through integer coordinates means and how it could help me in any way.

Answer 1

a) Factorization of P(x) is : P(x) = a(x-1)(x+2)(x-2).

So, you made it correctly.

b) You have said that graph of the function is provided to you. So, find the value of P(x) at x = -1 from the graph. Now, you have the value of P(-1).

Now, putting x = -1 in the function you get,

P(-1) = a(-1-1)(-1+2)(-1-2)

i.e., P(-1) = a*(-2)*1*(-3)

i.e., P(-1) = 6a

i.e., a = P(-1)/6

Since you have the value of P(-1), from above you will get the value of a, which is required.