Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Aim: How do we find the zeros of polynomial functions? Do Now: A rectangular playing field with

Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig.

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?


General Features of a Polynomial Function

Describe some basiccharacteristics of thispolynomial function:

4

2

-2

-4

-6

-8

5

Continuousno breaks in curve

Smoothno sharp turns

10

8

6

4

2

-2

5

discontinuous4

2

-2

-4

-6

5

sharp turn

NOT POLYNOMIAL FUNCTIONS

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.


General Features of a Polynomial Function

3 2( ) 2 5 2 5P x x x x

Leading Coefficient

Cubic term

Quadratic term

Linear term

Constant term

Standard FormDegree

Polynomial of 4 terms


General Features of a Polynomial FunctionSimplest form of any polynomial:

y = xn n > 0

6

4

2

-5 5

f x = x26

4

2

-5 5

h x = x4

6

4

2

-5

q x = x6

When n is evenlooks similar to x2

When n is oddlooks similar to x3

4

2

-2

-4

-5 5

r x = x3

-5

4

2

-2

-4

s x = x5

-5

4

2

-2

-4

t x = x7

The greater the value of n, the flatter thegraph is on the interval [ -1, 1].


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

-5 5

4

2

-2

u x = x4

5

4

2

-2v x = x-4 4-2


Zeros of Polynomial Functions

The zero of a function is a number x for which f(x) = 0. Graphically it’s the pointwhere the graph crosses the x-axis.

Ex. Find the zeros of f(x) = x2 + 3x

f(x) = 0 = x2 + 3x = x(x + 3)

x = 0 and x = -3

4

2

-2

-5

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

How many roots does f(x) = x2 + 1 have?

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


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

Function Zeros

1stn = 1

f(x) = x – 3 x = 3

2ndn = 2

f(x) = x2 – 6x + 9 = (x – 3)(x – 3)

x = 3 andx = 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


Finding Zeros

Find the zeros of f(x) = x3 – x2 – 2x

f(x) = 0 = x3 – x2 – 2x = x(x2 – x – 2) = x(x – 2)(x + 1)

x = 0, x = 2 and x = -1

2

-2

-4

•Has at most 3 real roots

•Has 2 relative extrema


Finding a Function Given the Zeros

Write a quadratic function whose zeros (roots) are -2 and 4.

x = -2 x = 4

x + 2 = 0 x – 4 = 0reverse the processused to solve the quadratic equation.(x + 2)(x – 4) = 0

x2 – 2x – 8 = 0x2 – 2x – 8 = f(x)

Find a polynomial function with the followingzeros: -2, -1, 1, 2

f(x) = (x + 2)(x + 1)(x – 1)(x – 2) f(x) = (x2 – 4)(x2 – 1) = x4 – 5x2 + 4


Multiplicity

Find the zeros of f(x) = x4 + 6x3 + 8x2.

f(x) = x4 + 6x3 + 8x2

2 2 6 8 GCFx x x

2 4 2 Factor trinomialx x x

2 0

4 0 Zero Product Property

2 =0

x

x

x

0 Multiple zeros - Multiplicity of 2

4

2

x

x

x

6

4

2

-2

-5

f x = x4+6x3+8x2

A multiple zero has a multiplicity equal to the numbers of times the zero occurs.


Regents Prep

The graph of y = f(x)

is shown at right.

Which set lists all the real solutions of f(x) = 0?

1. {-3, 2}

2. {-2, 3}

3. {-3, 0, 2}

4. {-2, 0, 3}

15

10

5

-5

-10

-15

-2 2 4


Model Problem

Find the zeros of f(x) = 27x3 + 1.327 1 0 Sum of perfect cubesx

Factoring Difference/Sum of Perfect Cubes

u3 – v3 = (u – v)(u2 + uv + v2)

u3 + v3 = (u + v)(u2 – uv + v2)

3 33 1 0 Rewrite as sum of cubesx

23 1 3 3 1 0 Factorx x x

3 1 u x v

23 1 9 3 1 0 Simplifyx x x

2

3 1 0 Zero Product Property

9 3 1 0

x

x x

1

3 1 0; 3

x x 29 3 1 0

Quadratic Formula

x x


Model Problem

Find the zeros of f(x) = 27x3 + 1.1

3 1 0; 3

x x

29 3 1 0

Quadratic Formula

x x 2 4

2

b b acx

a

23 3 4 9 1

2 9x

3 27 3 3 3

18 18

i

1 3

6

i

1 1 3 and

3 6

ix


Polynomial in Quadratic Form4 2( ) 4 4f x x x Find the zeros

2 22 22 2 0x x

2

2( ) 4 4 2 2 0

u x

f u u u u u

2 2 0; 2u u u

= 0( ) 0

the zeros

of function

f x

22 ; 2u x x 22 ; 2u x x

2 u’s – 4 zeros

zero @ + 2 - Multiplicity of 2

zero @ 2 - Multiplicity of 2

22 2 0x 2

2

2 0; 2

2 0 2

x x

x x


Finding zeros by Factoring by Groups

f(x) = x3 – 2x2 – 3x + 6

Group terms (x3 – 2x2) – (3x – 6) = 0

Factor Groups x2(x – 2) – 3(x – 2) = 0

Distributive Property (x2 – 3)(x – 2) = 0

Find the roots of the following polynomial function.

x3 – 2x2 – 3x + 6 = 0

Solve for x x2 – 3 = 0; 3 1.73x x – 2 = 0; x = 2

6

4

2

f x = x3-2x2-3x +6


Regents Prep

Factored completely, the expression 12x4 + 10x3 – 12x2 is equivalent to

2

2 2

2

2

1) 4 6 3 2

2) 2 2 3 3 2

3) 2 2 3 3 2

4) 2 2 3 3 2

x x x

x x x x

x x x

x x x

Documents

Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Aim: How do we find the zeros of polynomial functions? Do Now: A rectangular playing field with