2.3 / 2.5 The Fundamental Theorem of Algebra
Review of Long Division
Review of Synthetic Division
The Fundamental Theorem of Algebra
and related theorems guarantee that for any polynomial function,
the degree of the polynomial equals the number of zeros.
For example, a 3rd degree polynomial has 3 zeros….a 4th degree polynomial has 4 zeros…and so on.
Tools for Finding Zeros of Polynomials:
I.
Factor Theorem: If a polynomial f(x) is divided by x – c, and the remainder f(c) = 0, then x –
c is a factor of f(x).
Use the Factor Theorem to determine if (x – 1) is a factor of
.
II.
Descartes’ Rule of Signs:
For a polynomial f(x),
1) The number of positive real zeros of f is either equal to the number of sign changes in f(x)
or less than that number by an even integer, and
2) The number of negative real zeros of f is either equal to the number of sign changes in f(-x)
or less than that number by an even integer.
3) The number of imaginary zeros + positive real zeros + negative real zeros will always
equal the degree of the polynomial.
EX: Determine the possible combinations of positive real, negative real and imaginary zeros.
a.) f (x) = 2x 3 + x 2 + x + 4
P
N
III.
I
4
3
2
b.) f (x) = x − 3x + 6x + 2x − 60
P
N
I
Rational Zero Test:
If the polynomial
has integer coefficients with
then any rational zero of f will be of the form p/q where p is a factor of
and q is a factor of
.
(Note: If the polynomial has irrational or imaginary zeros, you will NOT find them directly using the
Rational Zero Test.)
EX:
Determine the possible rational zeros.
f (x) = 2x 3 + x 2 + x + 4
Finding ALL roots of a polynomial:
1.
2.
3.
4.
Use Descartes’ Rule of Signs to determine the number of possible real zeros.
Use the Rational Zero Test to determine any rational roots that may exist.
Divide the original polynomial by factors associated with the known roots.
Factor further or use the quadratic formula to find remaining roots of the function.
EX: f (x) = x 3 − 7x − 6
EX:
EX: f (x) = x 4 + x 3 − 4 x 2 + 2x − 12
Note that in the examples above, the imaginary roots come in conjugate pairs.
EX: Find all zeros of
as linear factors.
given that 2 + 3i is a zero. Write your answer
Writing polynomial functions for a given set of zeros:
f(x) = (x – ROOT)(x – ROOT)(x – ROOT)…
EX: Determine a polynomial function with roots: 1, -2, 5
EX: Determine a polynomial functions with roots:
4, -1
EX: Determine a polynomial functions with roots: 1 + 2i, 1 – 2i, 3
EX: Determine a polynomial functions with roots: 6 – i, 4, -2