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GHSGT Worksheet!!

Polynomial Functions

2.1 (M3)

What is a Polynomial?

• 1 or more terms• Exponents are whole numbers (not a fractional)• Coefficients are all real numbers (no imaginary

#’s)• NO x’s in the denominator or under the radical

• It is in standard form when the exponents are written in descending order.

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

One Term:Monomial

Two Terms:Binomial

Three Terms:Trinomial

3+ Terms:Polynomial

Classify each polynomial by degree and by number 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

2

3 2) 7f

x x

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 + 31

SOLUTION

a. Yes it’s a Polynomial. It is in standard form.

Degree 4 - Quartic Its leading coefficient is 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.

SOLUTION

237)(. xxxgb

37)( 2 xxxg b. Yes it’s a Polynomial. Standard form is Degree 2 – Quadratic Leading Coefficient is

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.

GUIDED PRACTICE for Examples 1 and 2

Use direct substitution to evaluate the polynomial function for the given value of x.

4. f (x) = x4 + 2x3 + 3x2 – 7; x = –2

5

5. g(x) = x3 – 5x2 + 6x + 1; x = 4

9

Graph Trend based on Degree

• Even degree - end behavior going the same direction

• Odd degree – end behavior (tails) going in opposite directions

• Positive– Odd degree—right side up, left side down– Even degree—both sides up

• Negative– Odd degree—right side down, left side up– Even degree—both sides down

Graph Trend based on

Leading Coefficient

Symmetry: Even/Odd/Neither• First look at degree• Even if it is symmetric respect to y-axis

– When you substitute -1 in for x, none of the signs change

• Odd if it is symmetric with respect to the origin– When you substitute -1 in for x, all of the signs

change.

• Neither if it isn’t symmetric around the y axis or origin

Tell whether it is even/odd/neither

1. f(x)= x2 + 2

2. f(x)= x2 + 4x

3. f(x)= x3

4. f(x)= x3 + x

5. f(x)= x3 + 5x +1

Additional Vocabulary to Review• Domain: set of all possible x values• Range: set of all possible y values• Symmetry: even (across y), odd (around origin),

or neither• Interval of increase (where graph goes up to the

right)• Interval of decrease (where the graph goes

down to the right)• End Behavior: f(x) ____ as x+∞

f(x) ______ as x-∞

Math 3 Book

• Page 67 #1 – 4

• Page 68 #5 – 7

• Page 69 #1 – 5, 8 – 16• a) Classify by degree and # of terms• b) Even, Odd or Neither• c) End Behavior

Blue Algebra 2 Books

• P.429 # 4 –10

– #9 and 10• A) Domain and Range• B) Classify by degree and # of terms• C) Even, Odd or Neither• D) End