# Real Zeros of Polynomials

Document Sample

```					Real Zeros of
Polynomials
Section 3.4
College Algebra, MATH 171
Mr. Keltner
Fundamental
Theorem of Algebra
• The Fundamental Theorem of
Algebra, first proved by Carl Johann
Friedrich Gauss, tells us that:
– An nth-degree polynomial will have at most n
real zeros.
– Including the complex number system (i’s),
he notes that every nth-degree polynomial will
have precisely n real zeros.
– This leads into the Linear Factorization
Theorem, mentioned on pg. 298, which
incorporates the idea of complex zeros of
polynomial functions.
Zeros of a
Polynomial
• Zeros of a function are also known as:
– Roots
– X-intercepts (where y = 0)
• The zeros of a function may be
substituted into the original equation
and make it true.
• The degree of the equation (or
function) will be the MOST amount of
zeros we will have for our equation,
including complex zeros and those
factors with multiplicity.
A visual example
• Before we get to the        • From its factored form,
next theorem, look at         we see that P(x) has
an example of where it        zeros at 2, 3, and -4.
is applied.                 • When it is expanded (or
multiplied out), note
• Consider the polynomial       that the constant term
function                      of 24 will come from
P(x) = (x - 2)(x - 3)(x + 4)   multiplying (-2)(-3)(4).
• If we multiplied out its • This means the zeros of
factors, we get               the polynomial are all
P(x) = x3 - x2 - 14x + 24.    factors of the constant
term.
Rational Zeros
Theorem
• This theorem says that if we have a
polynomial function, like
P(x) = anxn + . . . + a1x + a0
and if this function has integer
coefficients, then each of its rational
zeros can be written in the form:
a
p factor of constant coefficient0

a
q factor of leading coefficient n
Example 1
• List the possible rational zeros of
the function using the Rational
Zero Test.
P(x) = 2x3 + x2 - 13x + 6
Must…find…zeros!

• Very similar to what we did in Section 3.3 if
• In this section, if we find one zero, it will help
us reduce the polynomial to find the others.
– This is so that the polynomial doesn’t look as large.
Example 2
• Find all real zeros of
P(x) = x4 - 5x3 - 5x2 + 23x + 10.
– Start by finding all the possible rational zeros of the
function that can be written in the form p/q.
– Once you have found a zero by synthetic division, you can
use the answer from that to try and factor or use synthetic
division again.
– You can always apply the Quadratic Formula if you have
difficulty factoring a 2nd-degree polynomial of the form ax2
+ bx + c, which can be set equal to zero and solved.
• To solve an equation of the form ax2 + bx + c = 0, use

b    b 2  4 ac
x
2a
Example 3
• Find all real zeros of
f(x) = 9x4 + 3x3 - 30x2 + 6x + 12.
Example 3:
Narrowing Down
Options
• The function f(x) = 9x4 + 3x3 - 30x2 + 6x + 12
has possible zeros at:
1, 1/3, 1/9, 2, 2/3, 2/9, 3, 3/3, 3/9, 4, 4/3,
4/9, 6, 6/3, 6/9, 12, 12/3, and 12/9.
• By narrowing down the values that are repeated
(or reduce so that they are equal to another
possible zero), we condense our list of possible
zeros to:
1, 1/3, 1/9, 2, 2/3, 2/9, 3, 4, 4/3, 4/9, 6,
and 12.
That’s almost half the amount we had before!
Example 3:
Narrowing Down MORE
Options
f(x) = 9x4 + 3x3 - 30x2 + 6x + 12
• Using the graph
QuickTime™ and a
PNG decompressor                 of f(x), what x-
are neede d to se e this picture.      value do you see
that is a root of
the function?
When we find one of our possible roots that works, we
can use synthetic division to help factor the rest of the
polynomial.
This can help us find the other roots of the function.
Complex Conjugates
Theorem
• When we investigate complex
conjugates, we will notice that
their product eliminates the i
terms.
• If our polynomial function has a
zero in the form a + bi, then its
conjugate, a - bi, is also a zero of
the function.
Example 4
• Find all real zeros of
f(x) = x3 - x + 6.
Package Design
• Package Design is an
occupation that deals with
polynomials regularly.
• This field of study strives
to make a package that
does two things:
– Uses as little material as
possible
– Holds as much of the
product as possible
Example 5:
Package Design
• From the diagram below for a Special K box
that is unfolded from an 18” by 36” piece of
cardboard, calculate the dimensions for a box
that maximizes its volume.
35-2x             35-2x
1”   x     2        x       2
x
2

18-x

x
2
Assessment
Pgs. 308-309:
#’s 5-70, multiples of 5

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 9 posted: 4/21/2012 language: pages: 16