Aim: How do we find the zeros of polynomial functions?

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Aim How do we find the zeros of polynomial
functions?

• Do Now

A rectangular playing field with a perimeter of
100 meters is to have an area of at least 500
square meters. Within what bounds must the
length of the rectangle lie?
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General Features of a Polynomial Function
DEFINITION Polynomial Function Let n be a
nonnegative integer and let a0, a1, a2, . . .
an-1, an be real numbers with an ? 0. The
function given by f(x) anxn an - 1xn - 1 .
. . . a2x2 a1x a0 is a polynomial function
of degree n. The leading coefficient is an. The
zero function f(x) 0 is a polynomial function.
It has no degree and no leading coefficient.
Describe some basic characteristics of
this polynomial function
Continuous
no breaks in curve
Smooth
no sharp turns
NOT POLYNOMIAL FUNCTIONS
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General Features of a Polynomial Function
Standard Form
Linear term
Polynomial of 4 terms
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General Features of a Polynomial Function
Simplest form of any polynomial y xn n gt 0
When n is odd
looks similar to x3
The greater the value of n, the flatter the graph
is on the interval -1, 1.
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Transformations of Higher Degree Polys
If k and h are positive numbers and f(x) is a
function, then f(x h) k shifts f(x) right
or left h units shifts f(x) up or down k units
f(x) (x h)3 k - cubic
f(x) (x h)4 k - quartic
ex. f(x) (x 4)4 2
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Zeros of Polynomial Functions
The zero of a function is a number x for
which f(x) 0. Graphically its the
point where the graph crosses the x-axis.

• For polynomial function f of degree n,
• the function f has at most n real zeros

• the graph of f has at most n 1 relative extrema
  (relative max. or min.).

f(x) 0 x2 3x x(x 3)
x 0 and x -3
How many roots does f(x) x2 1 have?
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Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where n
gt 0, then f has at least one zero in the complex
number plane.
Degree of polynomial Function Zeros
1st n 1 f(x) x 3 x 3
2nd n 2 f(x) x2 6x 9 (x 3)(x 3) x 3 and x 3
3rd n 3 f(x) x3 4x x(x 2i)(x 2i) x 0, x 2i, x -2i
4th n 4 f(x) x4 1 (x 1)(x 1)(x i)(x i) x 1, x -1, x i, x -i
repeated zero
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Finding Zeros
Find the zeros of f(x) x3 x2 2x

• Has at most 3 real roots

• Has 2 relative extrema

f(x) 0 x3 x2 2x x(x2 x 2) x(x
2)(x 1)
x 0, x 2 and x -1
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Finding a Function Given the Zeros
Write a quadratic function whose zeros (roots)
are -2 and 4.
reverse the process used to solve the quadratic
equation.
x 2 0
x 4 0
(x 2)(x 4) 0
x2 2x 8 0
x2 2x 8 f(x)
f(x) (x 2)(x 1)(x 1)(x 2)
x4 5x2 4
f(x) (x2 4)(x2 1)
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Multiplicity
Find the zeros of f(x) x4 6x3 8x2.
f(x) x4 6x3 8x2
A multiple zero has a multiplicity equal to the
numbers of times the zero occurs.
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Regents Prep

• The graph of y f(x)
• is shown at right.
• Which set lists all the real solutions of f(x)
  0?
• -3, 2
• -2, 3
• -3, 0, 2
• -2, 0, 3

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Model Problem
Find the zeros of f(x) 27x3 1.
Factoring Difference/Sum of Perfect Cubes u3 v3
(u v)(u2 uv v2) u3 v3 (u v)(u2 uv
v2)
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Model Problem
Find the zeros of f(x) 27x3 1.
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Polynomial in Quadratic Form
Find the zeros
0
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Finding zeros by Factoring by Groups
Find the roots of the following polynomial
function.
f(x) x3 2x2 3x 6
x3 2x2 3x 6 0
(x3 2x2) (3x 6) 0
Group terms
Factor Groups
x2(x 2) 3(x 2) 0
Distributive Property
(x2 3)(x 2) 0
x2 3 0
Solve for x
x 2 0 x 2
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Regents Prep
Factored completely, the expression 12x4 10x3
12x2 is equivalent to