MATH 1100
SECTION 5.3 Notes
Real Zeros of Polynomials – Text Pages 333-341
The Factor Theorem tells us that finding the zeros of a polynomial
is really the same thing as factoring it into linear factors.
Rational Zeros Theorem:
If the polynomial Px   a n x n  a n 1 x n 1  ...  a1 x  a0 has integer
p
coefficients, then every rational zero of P is of the form
where,
q
p is a factor of a0
and, q is a factor of an.
Example 1:
List all possible rational zeros given by the Rational Zeros Theorem:
Px   6 x 4  x 2  2 x  12
Guidelines for Finding the Rational Zeros of P(x):
1) List all possible rational zeros.
2) Use synthetic division to evaluate P(x) at each possible
rational zero. When the remainder is zero, note the quotient
you have obtained and move to step 3.
3) Repeat Step 1 and Step 2 for the quotient. Stop when you
reach a quotient that is quadratic or easily factored, then
find the remaining zeros using methods from Chapter 3.
Example 2:
Factor the polynomial Px   2 x 3  x 2  13x  6
Step 1: Find possible zeros using Rational Zeros Theorem:
Step 2: Check Possible Zeros For an Actual Zero:
Then, Use Synthetic Division to Factor:
Step 3: Factor Completely:
Example 3:
Factor the polynomial Px   2 x 3  3x 2  2 x  3
Example 4:
Factor completely Px   6 x 4  7 x 3  12 x 2  3x  2 .
UPPER BOUND & LOWER BOUND:
If every real root c of a polynomial equation satisfies a  c  b , then
we say that,
a is a lower bound
and, b is an upper bound
for the roots of the polynomial equation.
The Upper & Lower Bounds Theorem:
Let P be a polynomial with real coefficients.
1) If we divide P(x) by x  b (with b  0 ) using synthetic division, and if
the row that contains the quotient and remainder has no negative
entry, then b is an upper bound for the real zeros of P.
2) If we divide P(x) by x  a (with a  0 ) using synthetic division, and if
the row that contains the quotient and remainder has entries that
are alternately nonpositive and nonnegative, then a is a lower
bound for the real zeros of P.
[Note: Zero is considered positive or negative as needed for this theorem]
Example 5:
Show that all real zeros of the polynomial Px   x 4  3x 2  2 x  5 lie
between –3 and 2.
Variation in Sign:
If P(x) is a polynomial with real coefficients, written in descending
powers of x and omitting powers with coefficient of zero, then a variation
in sign occurs whenever adjacent coefficients have opposite signs.
Example 6:
Determine the number of variations in sign each polynomial has:
(a.)
x 2  4x  1
(b.)
2x3  x  6
(c.)
x 4  3x 2  x  4
Descarte’s Rule of Signs:
Let P(x) be a polynomial with real coefficients.
(1.)
(2.)
The number of positive real zeros of P(x) is either equal to the
number of variations in sign of P(x) or is less than that by an
even whole number.
The number of negative real zeros of P(x) is either equal to
the number of variations in sign in P(-x) or is less than that
by an even whole number.
Example 7:
Use Descarte’s Rule of Signs to determine the possible number of
positive and negative real zeros of the polynomial,
P x   3 x 6  4 x 5  3 x 3  x  3