Polynomial Functions
Polynomials of Higher Degree & Division
2.2 & 2.3
Polynomial Function
Let a0, a1, a2, …, an-1, an be real numbers with an ≠ 0,
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
is a polynomial function of x with degree n.
Polynomials are continuous with smooth rounded turns.
Leading Coefficient Test (LCT)
n even; an>0
n even; an<0
n odd; an>0
n odd; an<0
Real Zeros (Equivalent Statements)
1. x = a is a zero of the function f
2. x = a is a solution of the polynomial equation f(x) = 0
3. (x – a) is a factor of f(x)
4. (a, 0) is an x-intercept of the graph of f
Note: A polynomial function of degree of n, has at most n real
zeros and at most n-1 turning points.
Repeat Roots (Zeros)
  A factor (x – a)k, k > 1 yield a repeated zero x = a of
multiplicity of k.
  If k is odd, the graph crosses at x = a.
  If k is even, the graph touches the x-axis (but does not cross) at
x = a.
Intermediate Value Theorem (IVT)
Let a and b be real numbers such that a < b.
If f is a polynomial function such that f(a) ≠ f(b), then in the
interval [a, b] f takes every value between f(a) and f(b).
Factor Theorem:
A polynomial f(x) has a factor (x – k) iif f(k)=0.
Remainder Theorem:
If a polynomial f(x) is divided by x – k, the remainder is r = f(k).
Long Division always works!
Synthetic Division works only if the divisor is the form (x – k)