2.1 Evaluate and Graph Polynomial Functions

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2.1 Evaluate and Graph Polynomial Functions

• Objectives
• Identify, evaluate, add, and subtract polynomials
• Classify polynomials, and describe the shapes of
  their graphs

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What is a Polynomial?

• 1 or more terms
• Exponents are whole numbers (not a radical)
• Coefficients are all real numbers (no imaginary
  s)
• It is in Standard Form when the exponents are
  written in descending order.

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Definitions for Polynomials

• Monomial a numeral, variable, or the product of
  a numeral and one or more variables
• Ex
• Constant a monomial w/ no variables
• Ex
• Coefficient numerical factor in a monomial
• Ex
• Degree of a Monomial sum of exponents of its
  variables
• Ex See Below

NOT QUOTIENT! i.e. x cant be on bottom!!!!
Give the degree for the following
monomial. 4x2y3z _________ ab4c2 ________
8 ________
7
6
0
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Definitions for Polynomials

• Polynomial is many (more than 1) monomials
  connected by addition or subtraction.

(5x 4)
(2x2 3x 2)
Binomial - ___________ Trinomial - ______________

• Degree of the Polynomial
  
  is the degree of its highest monomial term

Example Give the degree of the
polynomial. 4x3 6x2 -8x5 6 ________
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Classification of a Polynomial
Degree Name Example






n 0
constant
3
linear
n 1
5x 4
quadratic
n 2
2x2 3x - 2
cubic
n 3
5x3 3x2 x 9
quartic
3x4 2x3 8x2 6x 5
n 4
-2x5 3x4 x3 3x2 2x 6
n 5
quintic
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Classify each polynomial by degree and by number
of terms.
a) 5x 2x3 2x2
b) x5 4x3 x5 3x2 4x3
c) x2 4 8x 2x3
d) 3x3 2x x3 6x5
e) 2x 5x7
quintic trinomial
cubic polynomial
Not a polynomial
cubic trinomial
quadratic monomial
7th degree binomial
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EXAMPLE 1
Identify polynomial functions
Decide whether the function is a polynomial
function.If so, write it in standard form and
state its degree, type, and leading coefficient.
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SOLUTION
a. The function is a polynomial function that
is already written in standard form. It
has degree 4 (quartic) and a leading
coefficient of 1.
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EXAMPLE 1
Identify polynomial functions
Decide whether the function is a polynomial
function.If so, write it in standard form and
state its degree, type, and leading coefficient.
SOLUTION
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EXAMPLE 1
Identify polynomial functions
Decide whether the function is a polynomial
function.If so, write it in standard form and
state its degree, type, and leading coefficient.
c. f (x) 5x2 3x 1 x
SOLUTION
c. The function is not a polynomial function
because the term 3x 1 has an exponent
that is not a whole number.
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EXAMPLE 1
Identify polynomial functions
Decide whether the function is a polynomial
function.If so, write it in standard form and
state its degree, type, and leading coefficient.
d. k (x) x 2x 0.6x5
SOLUTION
d. The function is not a polynomial function
because the term 2x does not have a
variable base and an exponent that is a
whole number.
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for Examples 1 and 2
GUIDED PRACTICE
Decide whether the function is a polynomial
function. If so, write it in standard form and
state its degree, type, and leading coefficient.
2. p (x) 9x4 5x 2 4
1. f (x) 13 2x
not a polynomial function
polynomial function f (x) 2x
13 degree 1, type
linear, leading coefficient 2
3. h (x) 6x2 p 3x
polynomial function h(x) 6x2 3x p
degree 2,
type quadratic,
leading coefficient 6
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EXAMPLE 2
Evaluate by direct substitution
Use direct substitution to evaluate f (x) 2x4
5x3 4x 8 when x 3.
f (x) 2x4 5x3 4x 8
Write original function.
f (3) 2(3)4 5(3)3 4(3) 8
Substitute 3 for x.
162 135 12 8
Evaluate powers and multiply.
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Simplify
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for Examples 1 and 2
GUIDED PRACTICE
Use direct substitution to evaluate the
polynomial function for the given value of x.
4. f (x) x4 2x3 3x2 7 x 2
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ANSWER
5. g(x) x3 5x2 6x 1 x 4
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ANSWER
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Solving by Synthetic Substitution (Division)
(x - 2) is a Factor of use x 2
Use the Polynomials coefficients
Drop 1st coefficient down
Multiply
Answer
Remainder if there is any
Add Down
The Solution starts with one degree less than
original
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Homework

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