Key Terms
Two factors determine it
(1) even or odd exponent, (2) positive or negative coefficient.
General form
F(x) = a_n*x^n + ... + a_2*x^2 + a_1*x + a_0
Degree
Highest exponent in the polynomial Leading term: the term with the highest exponent Leading coefficient: the coefficient
Example
F(x) = -3x^4 - 9x^3 + 12x^2 Leading term is -3x^4. Even degree, negative coefficient.
Turning point
Where the graph changes from increasing to decreasing or vice versa. These are local highs and lows.
KEY RULE
A polynomial of degree n has AT MOST n x-intercepts and AT MOST n - 1 turning points.
Points
(2, 0), (-1, 0), (4, 0)
PRINCIPLE OF ZERO PRODUCTS
If ab = 0, then a = 0 or b = 0. This is why factored form makes finding zeros easy.
Add end behavior
Same direction = even, opposite = odd.
Leading term if expanded
-2x^3. Odd degree, negative coefficient.
EXAMPLE
Degree 4 polynomial, real coefficients, zeros at -3, 2, and i. Conjugate pair adds -i as a fourth zero.
Local (relative) max or min
Highest or lowest point in an open interval around x = a. Not necessarily the highest/lowest point overall.
Global (absolute) max or min
Highest or lowest point on the entire graph.
Given polynomial f(x) and divisor d(x)
F(x) = d(x) * q(x) + r(x)
Use it
Instead of plugging k into f(x) by hand, run synthetic division. The last number is f(k).