The graph shows a cubic function: 1. how many real roots does the function have? a. 0 b. 1 c. 2 d. 4 e. 3 2. complete the equation of the graphed function. y= (x - + ? )^2 the choices are 0,3,1,2,4

The polynomial has three real roots (of which one counts twice). It's formula is

The roots of a polynomial are the points in which the polynomial inersects the x-axis. So, you might be tempted to say that there are two roots, and .

But now let's face the problem in an analytic way: if is a solution for the polynomial , then can be factored as where is some remainder.

So, since we are sure that and are solutions, we know that we can write our polynomial as

But since the polynomial is a cubic (i.e. degree 3), must be a polynomial of degree 1, i.e. for some

But then we would have

and so we would have a third solution , but we can see from the graph that the only solutions are and .

This means that is actually a copy of one of the two other factors, so that it doesn't add new solutions. So, we have two cases:

or

The answer lies in the shape of the polynomial: you can see that the graph is tangent to the x-axis in , whereas it crosses the x-axis at . This tells us that the double solution is , and thus the correct solution is

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step-by-step explanation:

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