Warm Up
• For the given polynomial, one zero is given. Find all
other remaining zeroes:
P(x) = x3 – x2 - 4x – 6; 3 is a zero
Warm Up
• Find a polynomial function P(x) of least possible degree,
having real coefficients, with the given zeroes. Assume
the leading coefficient to be 1.
Zeroes: 1 − 3, 1 +
3 and 1
Section 3.7, Day 2
Exploring the Rational Zeroes Theorem,
Descartes’ Rule of Signs
Properties of Polynomial
Equations
• If a polynomial has a degree of n, the equation has “n” roots.
– Real Roots = x – intercepts.
– Consider multiplicity
– Consider imaginary/complex roots.
• If a + bi is a root, then a – bi is also a root.
• Example: Find all roots for x4 – x3 + 7x2 – 9x - 18
The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where
n > 1, then the equation f(x) = 0 has at
least one complex root.
Remember that having roots of 3, -2, etc. are
complex roots because 3 can be written 3+0i
and -2 can be written as -2+0i.
The Linear Factorization Theorem
• An nth degree polynomial can be expressed as the
product of a non-zero constant and n linear factors.
– Unless you know the constant, use “k” in front of the
polynomial.
Find a fourth-degree polynomial function f(x) with real
coefficients that has -1, 1/2 and i as zeros and such
that f(1)=- 4
Remember, if “i” is a factor, then….
Write out each factor.
F(x) = k(x
) (x
) (x
) (x
Rewrite ANY fractional factor (x
𝒂
– )
𝒃
)
as (bx – a)
The Linear Factorization
Theorem, cont….
• Replace F(x) with ______ and x with
_____ to find “k”
• Multiply out to find the polynomial in
standard form.
The Rational Zero Theorem
The Rational Zero Theorem gives a list of possible rational zeros of a
polynomial function. Equivalently, the theorem gives all possible rational roots
of a polynomial equation. Not every number in the list will be a zero of the
function, but every rational zero of the polynomial function will appear
somewhere in the list.
The Rational Zero Theorem
p
If f(x) = anxn + an-1xn-1 +…+ a1x + a0 has integer coefficients and
q
p
(where is reduced) is a rational zero, then p is a factor of the constant
q
term a0 and q is a factor of the leading coefficient an.
EXAMPLE: Using the Rational Zero Theorem
List all possible rational zeros of f(x) = 15x3 + 14x2 - 3x – 2.
Solution
The constant term is –2 and the leading coefficient is 15.
Factors of the constant term, - 2
Factors of the leading coefficient, 15
1,  2
=
1,  3,  5,  15
Possible rational zeros =
= 1,  2,
 13 ,  23 ,
Divide 1
and 2
by 1.
Divide 1
and 2
by 3.
 15 ,  52 ,
Divide 1
and 2
by 5.
1 ,  2
 15
15
Divide 1
and 2
by 15.
There are 16 possible rational zeros. The actual solution set to f(x) = 15x3 +
14x2 - 3x – 2 = 0 is {-1, -1/3, 2/5}, which contains 3 of the 16 possible solutions.
Example
List all possible rational zeros of f(x)=x3-3x2-4x+12
Find one of the zeros of the function using synthetic division, then
factor the remaining polynomial. What are all of the zeros of the
function? How can the graph below help you find the zeros?
Example
List all possible rational zeros of f(x)=6x3-19x2+2x+3
Starting with the integers, find one zero of the function using
synthetic division, then factor the remaining polynomial. What are
all of the zeros of the function?
Example
List all possible rational roots of x4-x3+7x2- 9x-18=0
Starting with the integers, find two roots of the equation using
synthetic division. The graph below will help you easily find those
roots. Factor the remaining polynomial. What are all of the roots of
the equation? The graph below will NOT help you find the
imaginary roots. Why?
Descartes' Rule of Signs
If f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 be a polynomial with real
coefficients.
1. The number of positive real zeros of f is either equal to the number
of sign changes of f(x) or is less than that number by an even integer.
If there is only one variation in sign, there is exactly one positive real
zero.
2. The number of negative real zeros of f is either equal to the number
of sign changes of f(-x) or is less than that number by an even
integer. If f(-x) has only one variation in sign, then f has exactly one
negative real zero.
EXAMPLE:
Using Descartes’ Rule of Signs
Determine the possible number of positive and negative real zeros of
f(x) = x3 + 2x2 + 5x + 4.
Solution
1. To find possibilities for positive real zeros, count the number of sign
changes in the equation for f(x). Because all the terms are positive, there
are no variations in sign. Thus, there are no positive real zeros.
2. To find possibilities for negative real zeros, count the number of sign
changes in the equation for f(-x). We obtain this equation by replacing x
with -x in the given function.
f(x) = x3 + 2x2 + 5x + 4
Replace x with -x.
f(-x) = (-x)3 + 2(-x)2 + 5(-x) + 4
= -x3 + 2x2 - 5x + 4
This is the given polynomial function.
EXAMPLE:
Using Descartes’ Rule of Signs
Determine the possible number of positive and negative real zeros of
f(x) = x3 + 2x2 + 5x + 4.
Solution
Now count the sign changes.
f(-x) = -x3 + 2x2 - 5x + 4
1
2
3
There are three variations in sign.
# of negative real zeros of f is either equal to 3, or is less than this number by
an even integer.
This means that there are either 3 negative real zeros
or 3 - 2 = 1 negative real zero.
Example
For f(x)=x3- 3x2- x+3 how many positive and
negative zeros are there? What are the
zeros of the function?
Example
For f(x)=x3- x2+4x- 4 how many positive and
negative zeros are there? Use a graphing
utility to find one real zero of the function.
What are all the zeros of the function?
p. 235, #65
• Use the given zero to completely factor
P(x) into linear factors
• Zero: i
• P(x) = x5 – x4 + 5x3 – 5x2 + 4x - 4
Putting it all together
• For the listed polynomial function:
a. Use Descartes’ Rule of Signs to find the possible number of
positive and negative real zeroes.
b. Use the rational zeroes theorem to determine the possible rational
zeroes of the function.
c.
Find the rational zeroes, if any.
d. Find any other real zeroes (those that CANNOT be written as a
fraction).
e. Find any other non-real complex zeroes, if any.
f. Find the x-intercepts of the graph, if any.
g. Find the y-intercept of the graph.
h. Use synthetic division to find P(4) and tell what that means.
i.
Sketch the graph and give its end behavior.
P(x) = -2x4 – x3 + x + 2
Closure
List all possible rational zeros of the
function f(x)=x3+3x2- 6x-8.
1
2
(b) 1, 2, 4, 8
(c)  1 , 2, 4,
2
(d) 1,
(a)  , 1, 2, 4, 8
Closure, cont….
What are the zeros of the function
f(x)=x3+2x2+8x+16? Find the first zero
using a graphing utility.
(a) -2, -2i 2, 2i 2
(b) 2, -2 2, 2 2
(c) 2, -2i, 2i
(d) -2, -2, 2