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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?
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
General Features of a Polynomial Function
DEFINITION:
Function
Describe somePolynomial
basic
characteristics
of thisinteger and let a , a ,
Let
n be a nonnegative
0
1
function:
a ,polynomial
. . . a , a be
real numbers with a ≠ 0.
4
2
2
n-1
n
n
The
function given by
Continuous
n - 1 + . . . . + a x2 + a x + a
f(x) =noanbreaks
xn + anin
x
curve
-1
2
1
0
5
-2
-4
is aSmooth
polynomial function of degree n. The
leading
coefficient
no sharp
turnsis an. The zero function
f(x) = 0 is a polynomial function. It has no
sharp turn
discontinuous
degree and no leading coefficient.
-6
-8
10
4
8
2
NOT POLYNOMIAL FUNCTIONS
6
5
4
-2
2
-4
5
-6
-2
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
General Features of a Polynomial Function
Standard Form
Degree
P( x) 2 x 5 x 2 x 5
3
2
Leading
Coefficient
Cubic Quadratic Linear Constant
term
term
term
term
Polynomial of 4 terms
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
General Features of a Polynomial Function
Simplest form of any polynomial:
y = xn
n>0
When n is even
When n is odd
looks similar to x2
looks similar to x3
f x = x2
r x = x 3
6
hx = x4
qx = x6
sx = x5
4
4
2
t x = x7
2
-5
-2
-5
5
-4
The greater the value of n, the flatter the
graph is on the interval [ -1, 1].
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
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
ux = x4
4
2
5
-5
vx = x-4 4 -2
-2
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Zeros of Polynomial Functions
The zero of a function is a number x for
which f(x) = 0. Graphically it’s 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.).
Ex. Find the zeros of
f(x) = x2 + 3x
f(x) = 0 = x2 + 3x = x(x + 3)
x = 0 and x = -3
4
2
-5
-2
How many roots does f(x) = x2 + 1 have?
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where
n > 0, then f has at least one zero in the
complex number plane.
Degree of
polynomial
1st
n=1
2nd
n=2
3rd
n=3
4th
n=4
Function
Zeros
f(x) = x – 3
x=3
f(x) = x2 – 6x + 9 =
(x – 3)(x – 3)
f(x) = x3 + 4x =
x(x – 2i)(x + 2i)
x = 3 and
x = 3 repeated zero
x = 0, x = 2i,
x = -2i
f(x) = x4 – 1 =
x = 1, x = -1,
(x – 1)(x + 1)(x – i)(x + i) x = i, x = -i
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Finding Zeros
2
Find the zeros of
f(x) = x3 – x2 – 2x
•Has at most 3 real roots
•Has 2 relative extrema
-2
f(x) = 0 = x3 – x2 – 2x =
x(x2 – x – 2) = x(x – 2)(x + 1)
x = 0, x = 2 and x = -1
Aim: Roots of Polynomial Equations
-4
Course: Alg. 2 & Trig.
Finding a Function Given the Zeros
Write a quadratic function whose
zeros (roots) are -2 and 4.
x=4
x = -2
x+2=0 x–4=0
(x + 2)(x – 4) = 0
x2 – 2x – 8 = 0
x2 – 2x – 8 = f(x)
reverse the process
used to solve the
quadratic equation.
Find a polynomial function with the following
zeros: -2, -1, 1, 2
f(x) = (x + 2)(x + 1)(x – 1)(x – 2)
f(x) = (x2 – 4)(x2 – 1) = x4 – 5x2 + 4
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Multiplicity
Find the zeros of f(x) = x4 + 6x3 + 8x2.
f(x) = x4 + 6x3 + 8x2
x2 x2 6 x 8
f x = x 4+6x 3+8x 2
6
GCF
x 2 x 4 x 2
4
Factor trinomial
2
x 4 0 Zero Product Property
x 2 =0
x 0 Multiple zeros - Multiplicity of 2
x2 0
-5
-2
x 4
x 2
A multiple zero has a multiplicity
equal to the numbers of times the
zero occurs.
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Regents Prep
The graph of y = f(x)
15
is shown at right.
10
Which set lists all
the real solutions
of f(x) = 0?
5
-2
2
1. {-3, 2}
-5
2. {-2, 3}
-10
3. {-3, 0, 2}
4. {-2, 0, 3}
Aim: Roots of Polynomial Equations
-15
Course: Alg. 2 & Trig.
4
Model Problem
Find the zeros of f(x) = 27x3 + 1.
27 x 3 1 0
3x
3
Sum of perfect cubes
1 0
Rewrite as sum of cubes
u 3x v 1
3
3 x 1 3 x
2
3x 1 0
Factor
3 x 1 9 x
2
3x 1 0
Simplify
3x 1 0
Zero Product Property
2
9 x 3 x 1 0
2
9
x
3x 1 0
1
3 x 1 0; x
3 Quadratic Formula
Factoring Difference/Sum of Perfect Cubes
2 + uv + v2)
u3 – v3 = (u – v)(u
Aim: Roots of Polynomial Equations
u3 + v3 = (u + v)(u2 – uv + v2)
Course: Alg. 2 & Trig.
Model Problem
Find the zeros of f(x) = 27x3 + 1.
1
3 x 1 0; x
3
9 x2 3 x 1 0
b b 2 4ac
x
Quadratic Formula
2a
x
3
3 4 9 1
2 9
2
3 27 3 3i 3 1 i 3
18
18
6
1
1 i 3
x and
3
6
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Polynomial in Quadratic Form
f ( x) 0
the zeros
of function
Find the zeros f ( x ) x 4 4 x 2 4 = 0
u x2
f ( u) u2 4u 4 u 2 u 2 0
u 2 u 2 0;
u2
u 2 x 2;
x 2
u 2 x 2;
x 2
2 u’s –
4 zeros
zero @ + 2 - Multiplicity of 2
zero @ 2 - Multiplicity of 2
x 2x
2
2
2
2 0 x 2
2
2
x 2 2 0;
x 2
x2 2 0
x 2
Aim: Roots of Polynomial Equations
2
0
Course: Alg. 2 & Trig.
Finding zeros by Factoring by Groups
Find the roots of the following
polynomial function.
f x = x 3-2x 2-3x +6
f(x) = x3 – 2x2 – 3x + 6 = 0
6
Group terms
(x3 – 2x2) – (3x – 6) = 0
Factor
Groups
x2(x – 2) – 3(x – 2) = 0
4
2
Distributive
Property
Solve for x
(x2 – 3)(x – 2) = 0
x2 – 3 = 0; x 3 1.73
x – 2 = 0;
x=2
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.
Regents Prep
Factored completely, the expression
12x4 + 10x3 – 12x2 is equivalent to
1) x 2 4 x 6 3 x 2
2) 2 2 x 2 3 x
3x2 2x
3) 2x 2 2 x 3 3 x 2
4) 2x 2 2 x 3 3 x 2
Aim: Roots of Polynomial Equations
Course: Alg. 2 & Trig.