Polynomial function(s): a polynomial function of degree n is a function of the form…
P(x) = a(n)x(n) + a(n-1)x(n-1) + … + a1x + a0
where n is a nonnegative integer and an =/= 0
Leading Term: of a polynomial is the term with the highest exponent.
Zeros of a polynomial: P(x) are numbers c, such that P© = 0 - “x coordinates of the x-intercepts.”
The end behavior of a polynomial: For the graph of any polynomial function, the ends will either point up or down. For example:
As x -> ∞; f(x) (will either) -> ∞ OR -∞
As x -> -∞; f(x) (will either) -> ∞ OR -∞
Odd Degree & Positive Leading Coefficient: Left down & Right up.
Even Degree & Positive Leading Coefficient: Both ends up.
Odd Degree & Negative Leading Coefficient: Left up & Right Down.
Even Degree & Negative Leading Coefficient: Both ends down.
The numbers a0, a1, a2, …, a(n)
are called coefficients of the Polynomial.
a0 is the constant coefficient or constant term.
The number a(n), the coefficient of the highest power, is the leading coefficient, and the term a(x)x(n) is the leading term.
If a polynomial consists of just a single term, then it’s called a monomial.
For example:
P(x) = x3 and Q(x) = -6x5 are monomials.
f(x) = a(1)x +/- a0 is considered a “Linear” equation.
f(x) = a(2)x2 +/- a(1)x +/- a0 is considered a “Quadratic” equation.
f(x) = a(3)x3 + a(2)x2 + a(1)x + a0 is considered a “Cubic” equation.
Even exponent: - Parabola - Bigger exponent means flatter base (near zero)(?) Odd exponent: - Zig-Zag(?) - Bigger exponent means flatter base (near zero)(?)