College Algebra: Lesson 5.3 BONUS Real Zeros of Polynomial Functions
Long Division
 Can be used to divide ANY polynomials, but it is longer than synthetic division.
 We can rewrite the quotient of two polynomial using the Division Algorithm: If f(x) and d(x) are
polynomials such that d ( x )  0 , and the degree of d(x) is less than or equal to the degree of f(x), there
exist unique polynomials q(x) and r(x) such that: f(x) = d(x)q(x)+ r(x) where r ( x)  0 or the degree of
r(x) is less than the degree of d(x). If the remainder r(x) is zero, d(x) divides evenly into f(x).
 This means we can rewrite the quotient as a product of the divisor and some polynomial plus the
f ( x)
r ( x)
remainder:
 q ( x) 
d ( x)
d ( x)
f ( x)
 If the degree of f(x) is greater than or equal to the degree of d(x), then the ratio
is improper. The
d ( x)
r ( x)
ratio
is called proper because the degree of r(x) is less than the degree of d(x).
d ( x)
Examples: Divide using long division.
1. 3 x  1 3 x 3  5 x 2  10 x  3
Synthetic Division
 Synthetic division is a “shortcut” for long division, but you can only divide by polynomials of the form
( x – k ).
 The same rules for rewriting the ratio of polynomials applies.
Examples: Divide using synthetic division.
1. x  1 x 2  5 x  6
The Rational Zero Test
 For any polynomial function f ( x)  a n x n  a n 1 x n 1      a 2 x 2  a1 x1  a0 with integer coefficients, the
p
possible rational zeros of f(x) of the form
where p is a factor of the constant term ( a 0 ) and q is a
q
factor of the leading coefficient, a n .
 Possible rational zeros = factors of constant term
factors of leading coefficient
 There are other IRRATIONAL ZEROS possible!
 The number of zeros of the polynomial is n...that includes all real and imaginary zeros!
Examples:
Step 1: List all of the potential rational zeros, separated by commas. Step 2: Use polynomial division and the
quadratic formula, if necessary, to identify the actual zeros.
1.) p( x)  4 x 4  28x 3  47 x 2  7 x  12
2.) H ( x)  9 x 3  13x  4
3.) G( x)  2 x 3  9 x 2  6 x  5
4. Using the rational zero theorem, list all of the solutions of the following polynomial, separated by commas.
 x  10 x 2  10  x 3  0
Descarte’s Rule of Signs
Let f ( x)  a n x n  a n 1 x n 1      a 2 x 2  a1 x1  a0 be a polynomial with real coefficients and a0  0 .
 The number of positive real zeros of f is either EQUAL TO the number of variations in sign of f(x) or
LESS THAN THAT NUMBER by an even integer.
 The number of negative real zeros of f is either EQUAL TO the number of variations in sign of f(-x) or
LESS THAN THAT NUMBER by an even integer.
 A variation in sign means that two consecutive coefficients have opposite signs.
Examples: Use Descarte's Rule of Signs to determine the possible number of positive and negative real zeros.
If there are multiple possibilities, please separate your answers with commas.
1. f ( x)  14 x10  9 x 8  9 x 6  10 x 4  5
2. f ( x)  x 3  2 x 2  x  2
Upper and Lower Bounds
 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.
 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. (This means there are no real zeros larger than c.)
 If c < 0 AND the numbers in the last row are alternately positive and negative (with zero entries being
either positive or negative), c is a lower bound for the real zeros of f. (This means there are no real
zeros smaller than c.)
Examples: Step 1 of 3: Use Descarte's Rule of Signs to determine the possible number of positive and negative
real zeros. If there are multiple possibilities, please separate your answers with commas. Step 2 of 3: Use
synthetic division to identity the best integer upper and lower bounds of the real zeros. (Although there may
be more than one correct bound, you must find the best integer value ---the smallest upper bound and largest
lower bound.) Step 3 of 3: Using your answers to the preceding steps, polynomial division, and the quadratic
formula, if necessary, list all of the zeros (including multiples of the same zero where applicable), separated by
commas.
1. g ( x)  x 3  6 x 2  25 x  150
2. F ( x)  x 3  2 x 2  x  2
The 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, there is at least one point c such that a  c  b and
f (c)  0 . That is to say that there is at least o.ne zero of f between a and b.
Examples: Given the function below, does the Intermediate Value Theorem guarantee that a real zero exists
between the indicated values:
5 3
,
1. g ( x)  6 x 4  390 x 3  858 x 2  1110 x  600 ;
2 2
2. g ( x)  6 x 4  78 x 3  6 x 2  78 x  6 ;
 7 1
,
6 6