Polynomials!!!

POLYNOMIAL Function:

• A polynomial is the monomial or the sum of monomials with all exponents as whole numbers and coefficients are all real numbers.

• Ex- x3+6x2+12x+8

Is it a polynomial?

a) 3 3f x x x

b)

c)

d)

4 23 2 5xf x x x x

4 16 2f x x x x

20.5 2f x x x

YES

NO

NO

YES

STANDARD FORM

The terms of a polynomial are in STANDARD FORM when:

• they are ordered from left to right in ______________ order; which means from the ____________ exponent to the ____________

decreasing largest

smallest

The _________exponent in the polynomial. It determines the _____________________.

Ex: The degree of –7x + 9 – 4x2 is ____ because ___ is the largest exponent in the polynomial.

largest

2

2

Classifying:

0 – Constant

1 – Linear

2 – Quadratic

3 – Cubic

4 - Quartic

Number of Zeros

DEGREE:

LEADING COEFFICIENT:

The ______________once in _________________.

CONSTANT:

The term _________________ a variable.

first coefficientstandard form

without

STANDARD FORM

a. Standard Form:

Degree:

Leading Coefficient:

Constant:

2 4 32x 4 3x 12x

4 3 23x 12x 2x 4 3

4,quartic 4

b. Standard Form:

Degree:

Leading Coefficient:

Constant:

3 x

x 3

1,linear

1

3

Some polynomials have SPECIAL NAMES that are determined by the following:

A. Their _________ or

B. Their _________ of terms

C. Look at the chart for an explanation.

degree

number

Polynomial # of Terms

Name by # of Terms

DEGREE Name by

Degree12

8x

4x2 + 3

5x3 + x2

3x2 – 4x + 6

7t4 – 7t + 3

1

1

2

2

3

MonomialMonomialBinomialBinomialTrinomial

0

1

2

3

2

ConstantLinear

Quadratic

Cubic

Quadratic3 Trinomi

al4 Quartic

How do we write in interval notation?

• x < 2….

• when you want include use a bracket [

• when you want to exclude use a parenthesis (

( , 2) Let’s draw a number line first….

Let’s do another type….4 9x

Draw a number line first….

4,9

Domain• all the x-values

• Read the graph from left to right

all the y-values Read the graph from bottom to top

Range

(2,4)

(-1,-5)

(4,0)

What is the domain of f(x)?

y = f(x)

Ex. 1

Domain

(2,4)

(-1,-5)

(4,0)

y = f(x)

Ex. 2: What is the range of f(x)?

Range

With polynomials….

• The DOMAIN is always All Reals,

• The RANGE will be:• All Reals,

• Lower Boundary to infinity,

• Negative infinity to Upper Boundary,

,

, LB, ,UB

Zeros/x-intercepts/Solutions/Roots

Where the graph

crosses the x-axis

What’s a zero?

x-interceptsWhere the graph crosses

the x-axis. Also called zeros.

1,0 & 5,0

Analyze the Graph of a Function

Zeros: 1, 5

X-Intercepts: (-2, 0) (-2, 0) (3,0)

Zeros, Roots:

x = -2, -2, 3

X-Intercepts – (-1,0)(1,0)(2,0)

Zeros, Roots

x = -1, 1, 2

y-intercepts

Where the graph crosses the y-axis

y-Intercept: (0,-12)

y-Intercept: (0,2)

Find the y-intercepts & number of zeros:

a)

b)

4 2f x 3x 5x 1

2f x 2x 3x 15

Y- int: (0, -1)

Y- int: (0, 15)

# of zeros: 4

# of zeros: 2

Find the following

1.Domain:

2.Range:

3. x-intercepts:

4. y-intercepts:

All reals

All reals

(-2,0)(-2,0)(1,0)

(0, -4)

Find the following

1.Domain:

2.Range:

3. Zeros:

4. y-intercepts:

All reals

[-4, ∞)

-2, 2

(0, -4) -5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5