5.3: LOCATING REAL ZEROS OF
POLYNOMIALS
College Algebra
RATIONAL ZERO THEOREM:
If f(x) = anxn + an-1xn-1 + . . . . + a1x1 + a0 is a
polynomial with integer coefficients, then any
rational zero of f must be of the form ,
where p is a factor of the constant term a0
and q is a factor of the leading coefficient an.
EX. 1: FOR EACH POLYNOMIAL THAT FOLLOWS,
LIST ALL THE POTENTIAL RATIONAL ZEROS.
a) = + − − EX. 1: FOR EACH POLYNOMIAL THAT FOLLOWS,
LIST ALL THE POTENTIAL RATIONAL ZEROS.
b) = − − − − DESCARTES’ RULES OF SIGNS:
The number of positive real zeros of f is either
the number of variations (changes) in sign of
f(x) or is less than this number by a positive
even integer.
The number of negative real zeros of f is
either the number of variations (changes) in
sign of f(-x) or is less than this number by a
positive even integer.
EX. 2: USE DESCARTES’ RULE OF SIGNS TO
DETERMINE THE NUMBER OF POSITIVE AND
NEGATIVE REAL ZEROS. THAN FIND ALL
ZEROS. (USE RATIONAL ZEROS THEOREM)
a) = + − − EX. 2: USE DESCARTES’ RULE OF SIGNS TO
DETERMINE THE NUMBER OF POSITIVE AND
NEGATIVE REAL ZEROS. THAN FIND ALL
ZEROS. (USE RATIONAL ZEROS THEOREM)
b) =
− + −
UPPER AND LOWER BOUNDS OF ZEROS:
Assume f(x) is a polynomial with real coefficients,
a positive leading coefficient, and a degree > 1.
Let ‘a’ and ‘b’ be fixed numbers with a < 0 < b.
Then:
1. No real zero of ‘f’ is larger than ‘b’ (we say ‘b’ is
an upper bound of the zeros of ‘f’) if the last
row in the synthetic division of f(x) by (x – b)
contains no negative numbers.
2. No real zero of ‘f’ is smaller than ‘a’ (we say ‘a’
is a lower bound of the zeros of ‘f’) if the last
row in the synthetic division of f(x) by (x – a)
has entries that alternate in sign.
(0 can be + or -)
EX. 3: USE SYNTHETIC DIVISION TO IDENTIFY
UPPER AND LOWER BOUNDS OF THE REAL
ZEROS OF THE FOLLOWING POLYNOMIAL:
= + − − EX. 4: USE DESCARTES’ RULES OF SIGNS,
RATIONAL ZERO THEOREM, AND UPPER/LOWER
BOUNDS OF ZEROS TO FIND ALL THE ZEROS OF
EACH POLYNOMIAL.
a) = − + + − EX. 4: USE DESCARTES’ RULES OF SIGNS,
RATIONAL ZERO THEOREM, AND UPPER/LOWER
BOUNDS OF ZEROS TO FIND ALL THE ZEROS OF
EACH POLYNOMIAL.
b) = − + − + INTERMEDIATE VALUE THEOREM:
Assume that f(x) is a polynomial with real
coefficients, and that ‘a’ and ‘b’ are real
numbers with a < b. If f(a) and f(b) differ in
sign, then there is at least one point ‘c’ such
that a < c < b and f(c) = 0. That is, at least one
zero of f lies between ‘a’ and ‘b.’
EX. 5: DECIDE WHETHER OR NOT THERE
MUST BE A SOLUTION BETWEEN THE REAL
INDICATED NUMBERS.
a) = + − ; EX. 5: DECIDE WHETHER OR NOT THERE
MUST BE A SOLUTION BETWEEN THE REAL
INDICATED NUMBERS.
b) = − + + + ;