Notes 2.3 / 2.5 The Fundamental Theorem of Algebra
Review of Long Division
x − 3 2 x 3 − 5x 2 + x − 8
Review of Synthetic Division
x − 3 2 x 3 − 5x 2 + x − 8
More Examples: Divide using either method where appropriate.
x 3 − 2x 2 + 3x + 7
x2 + 1
x3 + 5
x−2
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.
Determine the remaining factors of the polynomial and write your answer in factored form.
EX: f (x) = 3x 3 + 2x 2 − 19x + 6 given the factors : (x + 3)(x − 2)
Notes 2.5 (Day 2)
Tools for Finding Zeros of Polynomials:
Rational Zero Test:
If the polynomial f ( x) = a n x n + a n −1 x n −1 + ... + a1 x + a 0 has integer coefficients with
an ≠ 0 and a 0 ≠ 0
then any rational zero of f will be of the form p/q where p is a factor of ao and q is a factor of a n .
(Note: If the polynomial has irrational or imaginary zeros, you will NOT find them directly using the
Rational Zero Test.)
EX 1:
Determine the possible rational zeros.
f (x) = 2x 3 + x 2 + x + 4
Finding ALL roots of a polynomial:
1.
2.
3.
Use the Rational Zero Test to determine any rational roots that may exist. (p/q)
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 2: f (x) = x 3 − 7x − 6
EX 3: f (x) = 3x 3 − 4x 2 + 8x + 8
3
2
EX 4: f (x) = x + x − 2x + 12
2.5 (Day 3)
Solve for ALL roots of the polynomial.
EX 1: f ( x) = x 4 − 3 x 3 + x − 3
EX 2: 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 3: Find all zeros of f ( x) = x 4 − 4 x 3 + 12 x + 4 x − 13 given that 2 + 3i is a zero.
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:
3 ,− 3 , 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