Question

What does the degree of a polynomial expression tell you about its related polynomial function? Explain your thinking. Give an example of a polynomial expression of degree three. Provide information regarding the graph and zeros of the related polynomial function.

Answer 1

The graph of function like that may may never cross the x-axis,

so the function could have no real zeros.

If it does have zeros, they will come in pairs,

because "what goes up must come down" and vice versa.

Of course the function cannot have more zeros than its degree.

For example, a polynomial function of degree 6 could have 0, 2, 4, or 6 real zeros.

Answer 2

The polynomial degree relates to the number of "zeros" or points of intersection of the polynomial function (curve in the (x,y) plane) with the x axis (that is, points where y=0). These zeros can sometimes be coinciding but that phenomenon aside, you will see N such intercepts with the x axis for a polynomial expression of N-th degree.

Example of a polynomial of 3rd degree is: x^3 + 2 x^2 - x - 2

You can factor it to (x-1)(x+1)(x+2) to see that the zeros are +1, -1, and -2. The plot is attached as an image - note the intercepts. Lmk if you have questions.