Finding Zeros of a polynomial

Question

) A polynomial has 2 zeros.  

  a) Draw an example of a polynomial with 2 zeros      

       that has “left up, right down” end behavior.

Answer 1

In reality, we say a "left up, right down" has 3 zeros but we can make two of them the same. An example would be f(x) = -(x + 1)²(x - 1), or, multiplied out, f(x) = -x³ - x² + x + 1

Answer 2

1. The "zeros" of a polynomial are the point(s) where the graph crosses the x-axis.
2. The degree of a single-variable polynomial is the highest exponent of the variable, which tells us a LOT about the graph: the possible amount of zeros, the possible amount of turning points, and the end behaviors.
3. If the function has an even-degree, then the end behaviors go in the same direction (both up or both down). If the function has an odd-degree, then the end behaviors go in opposite directions ("left up, right down" or "left down, right up"). *So the curve in the question has an odd-degree.*
4. The degree of the polynomial is the most possible times the curve can cross the x-axis. All possible zeros is the degree minus 2. [ie: if the degree is 6, then the curve could cross the x-axis 6, 4, or 2 times. If a vertex is on the x-axis, then it counts twice bc it is a double-root.]

So this curve is not possible, if it is a polynomial. It cannot be an odd degree and cross the x-axis twice.