Section 18
The Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n, where n >0,
than f has at least one zero in the complex number system.
Linear Factorization Theorem: If f(x) is a polynomial of degree n, where n > 0, then f
has precisely n linear factors: f(x) = an(x – c1)(x – c2)…(x – cn) where c1,…,cn are
complex numbers.
The Rational Zero Test: If the polynomial f(x) = anxn + an-1xn-1 + …+a1x + a0 has
integer coefficients, every rational zero of f has the form:
Possible Rational Zeros 
p factors of a0

q factors of an
where p and q have no common factors other than 1.
Determine all the possible zeros by using the rational zero test.
Ex: f ( x)  2 x 3  4 x 2  x  3
Ex: f ( x)  7 x 3  19 x 2  33x  18
Complex Zeros Occur in Conjugate Pairs: Let f(x) be a polynomial function that has
real coefficients. If a +bi, where b ≠ 0, is a zero of the function, then the conjugate a – bi
is also a zero of the function.
Ex: Find a third degree polynomial with real coefficients that has -2 and 3 + i as zeros.
Factors of a Polynomial: Every polynomial of degree n > 0 with real coefficients can be
written as the product of linear and quadratic factors with real coefficients, where the
quadratic factors have no real zeros (imaginary).

A quadratic factor with no real zeros is said to be prime or irreducible over the
reals.
Ex: x 2  1  ( x  i)( x  i) is irreducible over the reals.
Ex: x 2  2  ( x  2 )( x  2 ) is reducible over the reals.
Ex: Find all zeros of f ( x)  x 4  3x 3  6 x 2  2 x  60 , given that 1 + 3i is a zeros of f.
Variation of Sign: 2 consecutive coefficients that have opposite signs
Descrates’ Rule of Signs: Let f(x) = anxn + an-1xn-1 + …+a1x + a0 be a polynomial with
real coefficients and a0 ≠ 0.
1. The number of positive real zeros of f is either equal to the number of variations
in the sign of f(x) of less than that number by an even integer.
2. The number of negative real zeros of f is either equal to the number of variations
in the sign of f(-x) or less than that number by an even integer
Ex: Determine the number of positive zeros and the number of negative zeros of
f ( x)  3 x 3  5 x 2  6 x  4 .
Upper and Lower Bound Rules: Let f(x) be a polynomial with real coefficients and a
positive leading coefficient. Suppose f(x) is divided by x – c, using synthetic division.
1. If c > 0 and each number in the last row is either positive or zero, c is an upper
bound for the real zeros of f.
2. If c < 0 and each number in the last row are alternatively positive and negative
[zero entries count as positive or negative], c is a lower bound for the real zeros of
f.
Ex: Use synthetic division to verify the upper and lower bounds of the real zeros:
f ( x)  x 4  4 x 3  16 x  16
a. Upper x = 5
b. Lower x = -3
Ex: Find all the zeros of the function and write the polynomial as a product of linear
factors:
f ( x)  x 3  6 x 2  13x  10
Ex: Write the polynomial: f ( x)  x 4  2 x 3  3x 2  12 x  18 [Hint: x2 – 6 is a factor]
a. as the product of factors that are irreducible over the rationals
b. as the product of linear and quadratic factors that are irreducible over the reals.
c. in completely factored form
Ex: Use the given zeros to find all the zeros of the function f ( x)  x 3  4 x 2  14 x  20
One zero is -1 – 3i.