2.1 Evaluate and Graph PolynomialFunctions2.1 Evaluate and Graph PolynomialFunctions
2.1 Evaluate and Graph PolynomialFunctions2.1 Evaluate and Graph PolynomialFunctions
Objectives:
•Identify, evaluate, add, and subtract polynomials
•Classify polynomials, and describe the shapes of theirgraphs
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 theexponents are written in descendingorder.
Definitions for Polynomials
•Monomial: a numeral, variable, or the product of anumeral 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
Give the degree for the following monomial.
4x2y3z _________ ab4c2 ________
8 ________
NOTQUOTIENT!i.e. “x” can’t beon bottom!!!!
6
7
0
•Polynomial: is many (more than 1) monomialsconnected by addition or subtraction.
Definitions for Polynomials
Binomial - ___________ Trinomial - ______________
• Degree of the Polynomial:is the degree of it’s highest monomial term
Example: Give the degree of the polynomial.
4x3 + 6x2 -8x5 – 6 ________
(5x + 4)
(2x2 + 3x – 2)
Classification of a Polynomial
Degree
Name
Example
-2x5 + 3x4 – x3 + 3x2 – 2x + 6
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5
constant
3
linear
5x + 4
quadratic
2x2 + 3x - 2
cubic
5x3 + 3x2 – x + 9
quartic
3x4 – 2x3 + 8x2 – 6x + 5
quintic
Classify each polynomial by degree and bynumber of terms.
a) 5x + 2x3 – 2x2
cubic trinomial
b) x5 – 4x3 – x5 + 3x2 + 4x3
quadratic monomial
c) x2 + 4 – 8x – 2x3
d) 3x3 + 2x – x3 – 6x5
cubic polynomial
quintic trinomial
e) 2x + 5x7
7th degree binomial
Not a polynomial
EXAMPLE 1
Identify polynomial functions
4
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree,type, and leading coefficient.
a. h (x) = x4 – x2 + 3
1
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.
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.
b. g (x) = 7x – 3 + πx2
SOLUTION
b. The function is a polynomial function written as
g(x) = πx2 + 7x – 3 in standard form. It has
degree 2(quadratic) and a leading coefficient of π.
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.
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.
GUIDED PRACTICE
for Examples 1 and 2
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree,type, and leading coefficient.
1. f (x) = 13 – 2x
polynomial function;f (x) = –2x + 13;degree 1,type: linear,leading coefficient: –2
2. p (x) = 9x4 – 5x – 2 + 4
3. h (x) = 6x2 + π – 3x
polynomial function;
h(x) = 6x2 – 3x + π ;degree 2,type: quadratic,leading coefficient: 6
not a polynomial function
EXAMPLE 2
Evaluate by direct substitution
Use direct substitution to evaluatef (x) = 2x4 – 5x3 – 4x + 8 when x = 3.
f (x) = 2x4 – 5x3 – 4x + 8
f (3) = 2(3)4 – 5(3)3 – 4(3) + 8
= 162 – 135 – 12 + 8
= 23
Write original function.
Substitute 3 for x.
Evaluate powers and multiply.
Simplify
GUIDED PRACTICE
for Examples 1 and 2
Use direct substitution to evaluate the polynomialfunction for the given value of x.
4. f (x) = x4 + 2x3 + 3x2 – 7; x = –2
5
ANSWER
5. g(x) = x3 – 5x2 + 6x + 1; x = 4
9
ANSWER
Solving by Synthetic Substitution (Division)
(x - 2) is a Factor ofuse x = 2
Use the Polynomials coefficients
Multiply
Answer
Add Down
Drop 1stcoefficientdown
Remainderif there isany
The Solution starts with onedegree less than original
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