Let f(x) be a polynomial function.

1. A real or complex (imaginary) number a is a zero of f(x) just in case f(a) = 0.

2. A zero of f(x) is an x-intercept of the graph of f(x) just in case it is a real number.

3. End Behavior of f(x) having degree at least 1.

Case 1. f(x) has End Behavior = Up_Up just in case:
the highest degree term of f(x) has an even exponent and a positive coefficient (e.g. like x^2, x^4).

Case 2. f(x) has End Behavior = Down_Down just in case:
the highest degree term of f(x) has an even exponent and a negative coefficient (e.g. like -x^2, -x^4).

Case 3. f(x) has End Behavior = Down_Up just in case:
the highest degree term of f(x) has an odd exponent and a positive coefficient (e.g. like x, x^3).

Case 4. f(x) has End Behavior = Up_Down just in case:
the highest degree term of f(x) has an odd exponent and a negative coefficient (e.g. like -x, -x^3).

4. Combining x-intercepts with peaks, valleys, plateaus.

Case 1. If f(x) has a multiple real zero at a, which occurs an even number of times, then f(x) has both an x-intercept and either a peak or a valley at x = a, e.g. (x -a)^2. In this case, the graph of f(x) "bounces off the x-axis at x = a."

Case 2. If f(x) has a multiple real zero at a, which occurs an odd number of times, then f(x) has both an x-intercept and a plateau at x = a, e.g. (x -a)^3. In this case, the graph of f(x) "pushes through the x-axis at x = a."

5. Factoring.

6. Quadratic Formula.

7. Zeroes for difference/sum of two cubes in 5 (above). Use the quadratic formula in 6. to show that if d is a nonzero real number, then the following polynomial functions have complex (imaginary) zeroes.

8. Special 4th degree f(x).