Section 4.3 and 4.4 Factoring Polynomial Functions
Today:
4.3 and 4.4
Announcements
 Homework 4.3 due Monday
 Homework 4.4 due Monday
 Quiz Monday on graphing a function when
it is factored
Polynomials are easiest to graph from factored from. From the
factored form, we get:



all the zeros
information about sign changes
even the leading term
Example: f ( x)   xx  1x  2x  3
Much of algebra is about putting polynomials in factored form.
Tools For Factoring
Fund Thm of Algebra
We can Find any Rational Root
We can factor any poly of degree 2.
Given one irrational or complex root, we can find another.
Fundamental Theorem of Algebra: A polynomial of degree n will have precisely n
zeros, and can be factored into n linear factors.
y = anxn + an-1xn-1 + a1x +a0 = k (x-r1)(x-r2) … (x-rn)
Examples:
(a) f ( x)  x 2  4 is degree 2, must have 2 zeros: f ( x)  ( x  2)( x  2)
(b) f ( x)  x3  2 x2  x  2 is degree 3, must have 3 zeros:
f ( x)  ( x  2)( x 2  1)  ( x  2)( x  1)( x  1)
Section 4.3 and 4.4
Factors and Zeros of Polynomials, and Finding Them
There are three ways to think about the zeros of a function:
way 1: a number x  r1 is a zero if you plug it into the function and get zero:
Example: x=2 is a zero of f ( x)  x3  2 x2  x  2 because:
f (2)  23  2  22  2  2  8  8  2  2  0
way 2: a number x  r1 is a zero if it intersects the x-axis there
Example: the graph
of f ( x)  x3  2 x2  x  2 shows that
x=2 is a zero
way 3: a number x  r1 is a zero if x  r1  factors into the function with no remainder
Example: x=2 is a zero is of f ( x)  x3  2 x2  x  2 because f ( x)  ( x  2)( x 2  1)
Check: f ( x)  ( x  2)( x 2  1)  x 3  x  2 x 2  2  x 3  2 x 2  x  2
Long Division of a Polynomal
is like division of numbers, but easier.
The factor Theorem
If c is a zero of a polynomial P(x), then:
 P(c)=0
 (x-c) is a factor of the polynomial P(x) (i.e., no remainder; R(x)=0)
 P(x)=(x-c)*Q(x)
Example:
P(x) = x2-4 =(x-2)(x+2)
x=2 is a zero so



P(2)=0: P(2) = 22-4 =0
(x-2) is a factor of the polynomial P(x) (i.e., no remainder; R(x)=0)
long division shows that Q(x) = (x+2)
P(x)=(x-2)*Q(x) = (x-2)*(x+2)
Rational Zero Theorem
In this section, we will learn how to find the zeros of a polynomial as long as they are
rational. (finding non-rational roots for another section)
The Rational Zero Test
Suppose that P( x)  a n x n  a n 1 x n 1    a 2 x 2  a1 x  a0 .
Then any rational root of P(x) will be p / q where p divides the constant term a0 and q
divides the leading coefficient an.
Example: f ( x)  x3  2 x2  x  2 has 3 zeros (x=2,1,-1). Show that these roots satisfy
the rational zero test.
a0 = 2, an = 1
x=2: 2 = p/q where p divides 2 and q divides 1
x=1: 1 = p/q where p divides 2 and q divides 1
x=-1: -1 = p/q where p divides 2 and q divides 1
(yes: p=2 and q=1)
(yes: p=1 and q=1)
(yes: p=-1 and q=1)
Worksheet Determine all possibilities for rational zeros.
f ( x)  x 4  2 x 3  x 2  3 x  4  0
a0 = 4
an = 1
possible zeros are p/q where p divides 4 and q divides 1
p = numbers that divide 4
q = numbers that divide 1
all possible
factors of 4  1,2,4
p
=
=
= ±1, ±2, ±4
1
factors of 1
q
Worksheet Determine all possibilities for rational zeros.
f ( x)  6 x 5  24 x 4  40 x 3  48x 2  24 x  4  0
a0 = -4, an = 6
all possible zeros =
all possible p/q =
factors of 4
 1,2,4
p
=
=
factors of 6  1,2,3,6
q
±1/±1, ±1/±2, ±1/±3, ±1/±6,
±2/±1, ±2/±2, ±2/±3, ±2/±6,
±4/±1, ±4/±2, ±4/±3, ±4/±6
= ±1, ±1/2, ±1/3, ±1/6
±2, ±2/3, ±4, ±4/3