Real Zeros of Polynomial Functions (1) Polynomial and Synthetic Division •  Remainder Theorem
•  Factor Theorem
Page 117
Long Division of Polynomials
Let’s do
the
division.
(the square root sign
is the closest thing to
the division box!)
original _ function(dividend) → f (x) = 2x 3 − 3x 2 + 5x + 2
divisor → D(x) = x + 3
quotient → Q(x) = 2x 2 − 9x + 32
−94
remainder →
x+3
Long Division of Polynomials with no remainder
Let’s do the division.
original _ function(dividend) → f (x) = x 3 − 3x 2 + 6x − 4
divisor → D(x) = x − 1
quotient → Q(x) = x 2 − 2x + 4
0
remainder →
=0
x −1
Long Division of Polynomials
You do the division!
original _ function(dividend) → f (x) = 3x 3 − 2x + 1
divisor → D(x) = x − 2
quotient → Q(x) = 3x 2 + 6x + 10
21
remainder →
x−2
NOTICE !!!!
Important!!!
Remainder Theorem:
If a polynomial, f (x), of degree n >= 1
is divided by x-c, then the remainder,
R= f (c).
Synthetic Division –
an easier way to divide
polynomials
Synthetic Division – Drop the first
term, the 3,
then multiply
by 2 and add
to next
column.
continue
Consider the
divisor of the
form x-c, write c
Evaluating a polynomial using synthetic
division.
Remainder, so f (4) = 193
Using Synthetic Division to find factors
It is if the division by -3 yields no remainder.
x – c form, so for x
+3, use -3
Remainder is
0, so it is a
factor
Important!!!!!
Factor Theorem:
•  If f (x) is a polynomial of degree n >=1 and
f (c) = 0, then x – c is a factor of f (x).
•  Conversely, if x – c is a factor of f (x), then
f (c) = 0.
Given a polynomial function, y = f (x), c is
said to be a ZERO of the function if f (c) = 0.
So . . .
f (-3) = 0
A second zero or root
of the function
Equivalent Statements
•  If f is a polynomial function and c is a real
number, then the following statements are
equivalent:
–  The number, c, is a zero of the function
–  The number, c, is a solution or root of the
equation, f (x) = 0
–  The polynomial, x – c, is a factor of f (x)
–  The number, c, is an x-intercept of the graph
of f
Multiplicity of a Zero:
Then we say that c is a ZERO OF
MULTIPLICITY s of the function f
-2 is a zero with multiplicity of 3, 1 is a zero with a
multiplicity of 4, and -1 is a zero with a multiplicity of 2
Even-Odd Multiplicity Rule:
If the multiplicity, s, is an even number, then the
polynomial function f touches the x-axis at x = c and
bounces back.
If the multiplicity, s, is an odd number, then the
polynomial function f crosses the x-axis at x = c.
x = 4, even
multiplicity
x = 2, odd
multiplicity
Synthetic Division for you to do . . . .
Answer:
No remainder, so x + 3 is a
factor
Problems for you . . .
#1-23 odds from page 127