P.o.D. – Factor each of the
following:
1.)  2 + 10 + 21
2.)  2 − 3 − 4
3.)  2 − 15 + 54
4.) 2 + 12 + 32
5.)  2 − 100
1.)
2.)
3.)
4.)
(x+3)(x+7)
(t-4)(t+1)
(x-6)(x-9)
(p+8)(p+4)
5.) (x+10)(x-10)
11-4: The Factor Theorem
Learning Target(s): I can find
zeros of polynomial functions
by factoring; apply the zeroproduct theorem and the
factor theorem; graph
polynomial functions;
estimate zeros using graphs.
Zero(s) of a Polynomial:
- Places where the graph
crosses the x-axis
- Solutions to the equation
- Also known as a root
Zero Product Theorem:
For all a and b, ab=0 if and
only if a=0 or b=0.
EX: An open box has sides of
length x, (24-2x), and (24-2x).
Thus, its volume is given by
V(x)=x(24-2x)(24-2x). Find
the zeros of V.
The Zero-Product Theorem
says that in order for an
equation to equal zero, each of
its factors could equal zero.
X=0
24-2x=0 24-2x=0
24=2x
24=2x
12=x
12=x
Solutions: 0,12,12  12 is a
double root
Factor Theorem:
(x-r) is a factor of P(x) if and
only if P(r)=0. In other words,
r is a zero of P.
EX: Find the roots of ( ) =
 4 −  3 − 20 2 by factoring.
Begin by taking out a common
factor.
( ) =  2 ( 2 −  − 20)
Now factor the trinomial by
un-FOILing.
( ) =  2 ( − 5)( + 4)
Use the Zero-Product
Property to set each factor
equal to zero and solve.
X+4=0
 2 = 0 x-5=0
X=-4
 = +0, −0 x=5
The zeros are x=0,0,5,-4.
O is a double root.
EX: Find the roots of ( ) =
 4 − 14 2 + 45
Since this has 3 terms, we
should try to un-FOIL.
( ) = ( 2 − 9)( 2 − 5)
Notice that one of these
factors is the difference of
squares.
( ) = ( − 3)( + 3)( 2 − 5)
Apply the zero-product
property.
x-3=0
x=3
x+3=0
x=-3
2 − 5 = 0
2 = 5
 = ±√5
The roots are 3, -3, √5, −√5.
This is an example of the
Fundamental Theorem of
Algebra. The FTA states that
you will have the same
number of roots as the highest
exponent.
EX: A polynomial function p
with degree 4 and leading
coefficient 1 is graphed below.
Find the factors of p(x) and
use them to write a formula
for p(x).
This graph crosses the x-axis,
or has zeros, at -7, -4, 0, and 3.
Therefore, it has factors of
(x+7), (x+4), x, and (x-3).
Expand these factors to find
the polynomial.
( ) = ( + 7)( + 4)( − 3)
= ( 2 + 7)( + 4)( − 3)
= ( 3 + 4 2 + 7 2 + 28)( − 3)
= ( 3 + 11 2 + 28)( − 3)
=  4 + 11 3 + 28 2 − 3 3 − 33 2 − 84
=  4 + 8 3 − 5 2 − 84
EX: Find the equation of a
polynomial whose zeros are -4,
7/2, and 5/3.
The factors are (x+4), (x-7/2),
and (x-5/3).
7
5
( ) = ( + 4) ( − ) ( − )
2
3
Or ( ) = ( + 4)(2 − 7)(3 − 5)
7
5
= ( −  + 4 − 14) ( − )
2
3
1
5
2
= ( +  − 14) ( − )
2
3
2
1 2
5 2 5
70
=  +  − 14 −  −  +
2
3
6
3
3
7 2 89
70
= −  − +
6
6
3
Or ( ) = 6 3 − 7 2 − 89 + 140
3
Upon completion of this
lesson, you should be able to:
1. Find zeros of a polynomial.
2. Find the roots of a
polynomial by factoring.
3. Use ZoomMath (not on
ACT).
For more information, visit
http://www.purplemath.com/modules/from
zero2.htm
HW Pg.756 2-12, 14-27
Quiz 11.1-11.4 tomorrow
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11.4 Notes (Completed) - Fort Thomas Independent Schools