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).